3865:
7352:
28243:
23029:
12497:
6342:
6834:
18909:
12022:
5887:
18444:
28440:
7347:{\displaystyle {\begin{aligned}{F_{n+1} \choose F_{n}}&=A^{n}{F_{1} \choose F_{0}}\\&=S\Lambda ^{n}S^{-1}{F_{1} \choose F_{0}}\\&=S{\begin{pmatrix}\varphi ^{n}&0\\0&(-\varphi )^{-n}\end{pmatrix}}S^{-1}{F_{1} \choose F_{0}}\\&={\begin{pmatrix}\varphi &-\varphi ^{-1}\\1&1\end{pmatrix}}{\begin{pmatrix}\varphi ^{n}&0\\0&(-\varphi )^{-n}\end{pmatrix}}{\frac {1}{\sqrt {5}}}{\begin{pmatrix}1&\varphi ^{-1}\\-1&\varphi \end{pmatrix}}{1 \choose 0},\end{aligned}}}
795:
14951:
21948:
21396:
945:
28749:
12492:{\displaystyle {\begin{aligned}1+&\varphi +\varphi ^{2}+\dots +\varphi ^{n}-\left(1+\psi +\psi ^{2}+\dots +\psi ^{n}\right)\\&={\frac {\varphi ^{n+1}-1}{\varphi -1}}-{\frac {\psi ^{n+1}-1}{\psi -1}}\\&={\frac {\varphi ^{n+1}-1}{-\psi }}-{\frac {\psi ^{n+1}-1}{-\varphi }}\\&={\frac {-\varphi ^{n+2}+\varphi +\psi ^{n+2}-\psi }{\varphi \psi }}\\&=\varphi ^{n+2}-\psi ^{n+2}-(\varphi -\psi )\\&={\sqrt {5}}(F_{n+2}-1)\\\end{aligned}}}
19300:
6337:{\displaystyle {\begin{aligned}{\vec {F}}_{n}&={\frac {1}{\sqrt {5}}}A^{n}{\vec {\mu }}-{\frac {1}{\sqrt {5}}}A^{n}{\vec {\nu }}\\&={\frac {1}{\sqrt {5}}}\varphi ^{n}{\vec {\mu }}-{\frac {1}{\sqrt {5}}}(-\varphi )^{-n}{\vec {\nu }}\\&={\cfrac {1}{\sqrt {5}}}\left({\cfrac {1+{\sqrt {5}}}{2}}\right)^{\!n}{\varphi \choose 1}\,-\,{\cfrac {1}{\sqrt {5}}}\left({\cfrac {1-{\sqrt {5}}}{2}}\right)^{\!n}{-\varphi ^{-1} \choose 1}.\end{aligned}}}
10862:
1053:
61:
28088:
14433:
22558:
188:
22791:
18904:{\displaystyle {\begin{aligned}{\bigl (}{\tfrac {2}{5}}{\bigr )}&=-1,&F_{3}&=2,&F_{2}&=1,\\{\bigl (}{\tfrac {3}{5}}{\bigr )}&=-1,&F_{4}&=3,&F_{3}&=2,\\{\bigl (}{\tfrac {5}{5}}{\bigr )}&=0,&F_{5}&=5,\\{\bigl (}{\tfrac {7}{5}}{\bigr )}&=-1,&F_{8}&=21,&F_{7}&=13,\\{\bigl (}{\tfrac {11}{5}}{\bigr )}&=+1,&F_{10}&=55,&F_{11}&=89.\end{aligned}}}
19018:
2234:
15874:
14946:{\displaystyle {\begin{aligned}(1-z-z^{2})s(z)&=\sum _{k=0}^{\infty }F_{k}z^{k}-\sum _{k=0}^{\infty }F_{k}z^{k+1}-\sum _{k=0}^{\infty }F_{k}z^{k+2}\\&=\sum _{k=0}^{\infty }F_{k}z^{k}-\sum _{k=1}^{\infty }F_{k-1}z^{k}-\sum _{k=2}^{\infty }F_{k-2}z^{k}\\&=0z^{0}+1z^{1}-0z^{1}+\sum _{k=2}^{\infty }(F_{k}-F_{k-1}-F_{k-2})z^{k}\\&=z,\end{aligned}}}
20283:
20668:
19898:
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8513:
837:. In the Sanskrit poetic tradition, there was interest in enumerating all patterns of long (L) syllables of 2 units duration, juxtaposed with short (S) syllables of 1 unit duration. Counting the different patterns of successive L and S with a given total duration results in the Fibonacci numbers: the number of patterns of duration
17538:
4320:
18304:
1924:
6821:
19295:{\displaystyle 5{F_{\frac {p\pm 1}{2}}}^{2}\equiv {\begin{cases}{\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {p}{5}}{\bigr )}\pm 5\right){\pmod {p}}&{\text{if }}p\equiv 1{\pmod {4}}\\{\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {p}{5}}{\bigr )}\mp 3\right){\pmod {p}}&{\text{if }}p\equiv 3{\pmod {4}}.\end{cases}}}
6557:
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16635:
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994:
population, assuming that: a newly born breeding pair of rabbits are put in a field; each breeding pair mates at the age of one month, and at the end of their second month they always produce another pair of rabbits; and rabbits never die, but continue breeding forever. Fibonacci posed the rabbit
22989:
Thus, a male bee always has one parent, and a female bee has two. If one traces the pedigree of any male bee (1 bee), he has 1 parent (1 bee), 2 grandparents, 3 great-grandparents, 5 great-great-grandparents, and so on. This sequence of numbers of parents is the
Fibonacci sequence. The number of
23659:
SadgurushiShya writes that
Pingala was a younger brother of PÄáčini . There is an alternative opinion that he was a maternal uncle of PÄáčini . ... Agrawala , after a careful investigation, in which he considered the views of earlier scholars, has concluded that PÄáčini lived between 480 and 410
17889:
23395:"For four, variations of meters of two three being mixed, five happens. For five, variations of two earlierâthree four, being mixed, eight is obtained. In this way, for six, of four of five being mixed, thirteen happens. And like that, variations of two earlier meters being mixed, seven
20099:
20487:
19717:
19323:
15660:
23317:. In particular, it is shown how a generalized Fibonacci sequence enters the control function of finite-horizon dynamic optimisation problems with one state and one control variable. The procedure is illustrated in an example often referred to as the BrockâMirman economic growth model.
18437:
16471:
2661:
8212:
23622:
Before
Fibonacci wrote his work, the sequence Fn had already been discussed by Indian scholars, who had long been interested in rhythmic patterns ... both Gopala (before 1135 AD) and Hemachandra (c. 1150) mentioned the numbers 1,2,3,5,8,13,21 explicitly " p. 100 (3d
21740:
8195:
1844:
23309:
or infinite series-parallel circuit) follow the
Fibonacci sequence. The intermediate results of adding the alternating series and parallel resistances yields fractions composed of consecutive Fibonacci numbers. The equivalent resistance of the entire circuit equals the golden
7823:
17354:
13733:
4061:
1578:
18147:
20846:
23894:
It is ironic that
Leonardo, who made valuable contributions to mathematics, is remembered today mainly because a 19th-century French number theorist, Ădouard Lucas... attached the name Fibonacci to a number sequence that appears in a trivial problem in Liber
23286:
factor 1.609344 for miles to kilometers is close to the golden ratio, the decomposition of distance in miles into a sum of
Fibonacci numbers becomes nearly the kilometer sum when the Fibonacci numbers are replaced by their successors. This method amounts to a
5876:
1228:
3103:
16115:
6691:
20464:
5332:
20076:
1398:
22917:, dividing the circle in the golden ratio. Because this ratio is irrational, no floret has a neighbor at exactly the same angle from the center, so the florets pack efficiently. Because the rational approximations to the golden ratio are of the form
19694:
15506:
6357:
16236:
25599:
En 1830, K. F. Schimper et A. Braun . Ils montraient que si l'on représente cet angle de divergence par une fraction reflétant le nombre de tours par feuille (), on tombe réguliÚrement sur un des nombres de la suite de
Fibonacci pour le numérateur
16482:
20882:
15112:
6682:
22963:, the distance from the center. Sunflowers and similar flowers most commonly have spirals of florets in clockwise and counter-clockwise directions in the amount of adjacent Fibonacci numbers, typically counted by the outermost range of radii.
21526:
5756:
13086:
4917:
2365:
17736:
2229:{\displaystyle {\begin{aligned}U_{n}&=a\varphi ^{n}+b\psi ^{n}\\&=a(\varphi ^{n-1}+\varphi ^{n-2})+b(\psi ^{n-1}+\psi ^{n-2})\\&=a\varphi ^{n-1}+b\psi ^{n-1}+a\varphi ^{n-2}+b\psi ^{n-2}\\&=U_{n-1}+U_{n-2}.\end{aligned}}}
2859:
14241:
3232:
22513:
The sequence of
Pythagorean triangles obtained from this formula has sides of lengths (3,4,5), (5,12,13), (16,30,34), (39,80,89), ... . The middle side of each of these triangles is the sum of the three sides of the preceding
16863:
15526:
11754:
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14056:
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5050:
3760:. For example, the initial values 3 and 2 generate the sequence 3, 2, 5, 7, 12, 19, 31, 50, 81, 131, 212, 343, 555, ... . The ratio of consecutive terms in this sequence shows the same convergence towards the golden ratio.
2963:
15966:
13272:
2537:
23152:. The maternal grandfather received his X chromosome from his mother, and the maternal grandmother received X chromosomes from both of her parents, so three great-grandparents contributed to the male descendant's X chromosome
7606:
23115:. The male's mother received one X chromosome from her mother (the son's maternal grandmother), and one from her father (the son's maternal grandfather), so two grandparents contributed to the male descendant's X chromosome
23032:
The number of possible ancestors on the X chromosome inheritance line at a given ancestral generation follows the
Fibonacci sequence. (After Hutchison, L. "Growing the Family Tree: The Power of DNA in Reconstructing Family
21531:
2444:
1290:
17265:
16329:
10250:
7995:
1710:
8217:
7686:
1929:
15869:{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{{F_{k}}^{2}}}={\frac {5}{24}}\!\left(\vartheta _{2}\!\left(0,{\frac {3-{\sqrt {5}}}{2}}\right)^{4}-\vartheta _{4}\!\left(0,{\frac {3-{\sqrt {5}}}{2}}\right)^{4}+1\right).}
22088:
The figure shows that 8 can be decomposed into 5 (the number of ways to climb 4 steps, followed by a single-step) plus 3 (the number of ways to climb 3 steps, followed by a double-step). The same reasoning is applied
21927:
7449:
5555:
2725:
23226:, etc. (This assumes that all ancestors of a given descendant are independent, but if any genealogy is traced far enough back in time, ancestors begin to appear on multiple lines of the genealogy, until eventually a
18997:
17032:
9557:
3837:
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22893:
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3647:
9283:
1446:
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15228:
15170:
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algorithm in which an unsorted list is divided into two lists whose lengths correspond to sequential
Fibonacci numbersâby dividing the list so that the two parts have lengths in the approximate proportion
20725:
11382:
5169:
20278:{\displaystyle {\bigl (}{\tfrac {13}{5}}{\bigr )}=-1:\qquad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {13}{5}}{\bigr )}-5\right)=-5,\quad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {13}{5}}{\bigr )}+5\right)=0.}
8000:
6585:
5476:
2542:
1715:
20663:{\displaystyle {\bigl (}{\tfrac {29}{5}}{\bigr )}=+1:\qquad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {29}{5}}{\bigr )}-5\right)=0,\quad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {29}{5}}{\bigr )}+5\right)=5.}
19893:{\displaystyle {\bigl (}{\tfrac {11}{5}}{\bigr )}=+1:\qquad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {11}{5}}{\bigr )}+3\right)=4,\quad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {11}{5}}{\bigr )}-3\right)=1.}
17677:
17607:
5763:
5404:
19505:{\displaystyle {\bigl (}{\tfrac {7}{5}}{\bigr )}=-1:\qquad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {7}{5}}{\bigr )}+3\right)=-1,\quad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {7}{5}}{\bigr )}-3\right)=-4.}
7681:. The matrix formed from successive convergents of any continued fraction has a determinant of +1 or â1. The matrix representation gives the following closed-form expression for the Fibonacci numbers:
10975:
1123:
12920:
11847:
2978:
22621:, and one with the fewest nodes for a given heightâthe "thinnest" AVL tree. These trees have a number of vertices that is a Fibonacci number minus one, an important fact in the analysis of AVL trees.
15981:
11917:
20340:
8508:{\displaystyle {\begin{aligned}F_{2n-1}&={F_{n}}^{2}+{F_{n-1}}^{2}\\F_{2n{\phantom {{}-1}}}&=(F_{n-1}+F_{n+1})F_{n}\\&=(2F_{n-1}+F_{n})F_{n}\\&=(2F_{n+1}-F_{n})F_{n}.\end{aligned}}}
5191:
19955:
15388:
12686:
11615:
10695:
1311:
7947:
3436:
3321:
20721:
19567:
18449:
14438:
12027:
6839:
5892:
4041:
2299:
18142:
16122:
10520:
10057:
9113:
2783:
11489:
10441:
925:
Variations of two earlier meters ... For example, for four, variations of meters of two three being mixed, five happens. ... In this way, the process should be followed in all
11235:
11094:
20336:
10359:
9842:
2528:
1915:
19951:
19563:
17533:{\displaystyle {\begin{cases}p=5&\Rightarrow p\mid F_{p},\\p\equiv \pm 1{\pmod {5}}&\Rightarrow p\mid F_{p-1},\\p\equiv \pm 2{\pmod {5}}&\Rightarrow p\mid F_{p+1}.\end{cases}}}
15018:
6580:
4461:
3273:
52:. The sequence commonly starts from 0 and 1, although some authors start the sequence from 1 and 1 or sometimes (as did Fibonacci) from 1 and 2. Starting from 0 and 1, the sequence begins
14381:
11993:
10918:
21418:
15288:
5648:
4315:{\displaystyle \varphi ^{n+1}=(F_{n}\varphi +F_{n-1})\varphi =F_{n}\varphi ^{2}+F_{n-1}\varphi =F_{n}(\varphi +1)+F_{n-1}\varphi =(F_{n}+F_{n-1})\varphi +F_{n}=F_{n+1}\varphi +F_{n}.}
3919:
3379:
12565:
11148:
10851:
9003:
4767:
966:
the indices from present to XII (months) as Latin ordinals and Roman numerals and the numbers (of rabbit pairs) as Hindu-Arabic numerals starting with 1, 2, 3, 5 and ending with 377.
18299:{\displaystyle \left({\frac {p}{5}}\right)={\begin{cases}0&{\text{if }}p=5\\1&{\text{if }}p\equiv \pm 1{\pmod {5}}\\-1&{\text{if }}p\equiv \pm 2{\pmod {5}}.\end{cases}}}
12924:
9953:
4809:
4549:
2290:
1695:
260:
9351:
3754:
482:
325:
24481:
1439:
15444:
10780:
2792:
936:(c. 1150) is credited with knowledge of the sequence as well, writing that "the sum of the last and the one before the last is the number ... of the next mÄtrÄ-váčtta."
14313:
14091:
8923:
4396:
4357:
16750:
8861:
11620:
9727:
3576:
16656:
13910:
13755:
8802:
417:
22353:, and a sum of Fibonacci numbers that satisfies these conditions is called a Zeckendorf representation. The Zeckendorf representation of a number can be used to derive its
21210:. All these sequences may be viewed as generalizations of the Fibonacci sequence. In particular, Binet's formula may be generalized to any sequence that is a solution of a
12529:
11533:
2888:
4798:
3948:
15881:
13114:
12017:
7483:
6816:{\displaystyle \Lambda ={\begin{pmatrix}\varphi &0\\0&-\varphi ^{-1}\!\end{pmatrix}},\quad S={\begin{pmatrix}\varphi &-\varphi ^{-1}\\1&1\end{pmatrix}}.}
22007:
8536:-th Fibonacci number from the closed-form matrix formula, but with fewer redundant steps if one avoids recomputing an already computed Fibonacci number (recursion with
24343:
14428:
10560:
4636:
4600:
3968:
3857:
23222:
23185:
23148:
23111:
23076:
23041:
inheritance line at a given ancestral generation also follows the Fibonacci sequence. A male individual has an X chromosome, which he received from his mother, and a
17113:
17080:
16908:
16644:
15006:
11268:
10286:
9675:
9642:
8739:
8701:
8668:
2374:
1237:
368:
10602:
1627:
1012:
At the end of the fourth month, the original pair has produced yet another new pair, and the pair born two months ago also produces their first pair, making 5 pairs.
21767:
14980:
10722:
9754:
9378:
8635:
8582:
6552:{\displaystyle F_{n}={\cfrac {1}{\sqrt {5}}}\left({\cfrac {1+{\sqrt {5}}}{2}}\right)^{\!n}-\,{\cfrac {1}{\sqrt {5}}}\left({\cfrac {1-{\sqrt {5}}}{2}}\right)^{\!n}.}
3703:
3676:
15439:
15321:
11411:
11029:
9870:
9609:
9583:
8608:
3108:
23534:
it was natural to consider the set of all sequences of and that have exactly m beats. ... there are exactly Fm+1 of them. For example the 21 sequences when
21959:
These numbers also give the solution to certain enumerative problems, the most common of which is that of counting the number of ways of writing a given number
16630:{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{F_{k}}}=\sum _{k=1}^{\infty }{\frac {1}{F_{2k-1}}}+\sum _{k=1}^{\infty }{\frac {1}{F_{2k}}}=3.359885666243\dots }
15410:
14266:
10995:
7359:
28631:
21097:{\displaystyle F_{1}=1,\ F_{3}=2,\ F_{5}=5,\ F_{7}=13,\ F_{9}={\color {Red}34}=2\cdot 17,\ F_{11}=89,\ F_{13}=233,\ F_{15}={\color {Red}610}=2\cdot 5\cdot 61.}
18922:
16928:
9383:
3555:. He wrote that "as 5 is to 8 so is 8 to 13, practically, and as 8 is to 13, so is 13 to 21 almost", and concluded that these ratios approach the golden ratio
22717:
head showing the arrangement in 21 (blue) and 13 (cyan) spirals. Such arrangements involving consecutive Fibonacci numbers appear in a wide variety of plants.
22375:
22828:
23333:
which uses Fibonacci intervals in some of its melodies; he viewed these as the musical counterpart to the elaborate harmony evident within nature. See also
13405:
13279:
5560:
4924:
3580:
26190:
25821:
24706:
Bugeaud, Y; Mignotte, M; Siksek, S (2006), "Classical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers",
24573:
4480:
1009:
At the end of the third month, the original pair produce a second pair, but the second pair only mate to gestate for a month, so there are 3 pairs in all.
17884:{\displaystyle {\begin{pmatrix}F_{m+1}&F_{m}\\F_{m}&F_{m-1}\end{pmatrix}}\equiv {\begin{pmatrix}1&1\\1&0\end{pmatrix}}^{m}{\pmod {n}}.}
12691:
17121:
16244:
10068:
5058:
21377:
Generating the next number by adding 3 numbers (tribonacci numbers), 4 numbers (tetranacci numbers), or more. The resulting sequences are known as
25819:
Patranabis, D.; Dana, S. K. (December 1985), "Single-shunt fault diagnosis through terminal attenuation measurement and using Fibonacci numbers",
25757:
17623:
17553:
5337:
15655:{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{F_{2k-1}}}={\frac {\sqrt {5}}{4}}\;\vartheta _{2}\!\left(0,{\frac {3-{\sqrt {5}}}{2}}\right)^{2},}
21841:
5494:
28274:
22813:
18432:{\displaystyle F_{p}\equiv \left({\frac {p}{5}}\right){\pmod {p}}\quad {\text{and}}\quad F_{p-\left({\frac {p}{5}}\right)}\equiv 0{\pmod {p}}.}
16466:{\displaystyle \textstyle {\frac {1}{F_{2k}}}={\sqrt {5}}\left({\frac {\psi ^{2k}}{1-\psi ^{2k}}}-{\frac {\psi ^{4k}}{1-\psi ^{4k}}}\right)\!.}
12815:
11758:
3766:
2673:
11859:
3868:
Successive tilings of the plane and a graph of approximations to the golden ratio calculated by dividing each Fibonacci number by the previous
28621:
2656:{\displaystyle {\begin{aligned}a&={\frac {U_{1}-U_{0}\psi }{\sqrt {5}}},\\b&={\frac {U_{0}\varphi -U_{1}}{\sqrt {5}}}.\end{aligned}}}
4645:
28714:
21211:
15331:
12592:
11538:
10618:
9119:
1095:
23247:, it may reflect off three surfaces: the top, middle, and bottom surfaces of the two plates. The number of different beam paths that have
7856:
3473:
25332:
21735:{\displaystyle {\frac {x}{1-x-x^{2}}}=x+x^{2}(1+x)+x^{3}(1+x)^{2}+\dots +x^{k+1}(1+x)^{k}+\dots =\sum \limits _{n=0}^{\infty }F_{n}x^{n}}
20672:
3981:
8190:{\displaystyle {\begin{aligned}{F_{m}}{F_{n}}+{F_{m-1}}{F_{n-1}}&=F_{m+n-1},\\F_{m}F_{n+1}+F_{m-1}F_{n}&=F_{m+n}.\end{aligned}}}
18001:
Fibonacci numbers. In 2006, Y. Bugeaud, M. Mignotte, and S. Siksek proved that 8 and 144 are the only such non-trivial perfect powers.
10446:
1839:{\displaystyle {\begin{aligned}\varphi ^{n}&=\varphi ^{n-1}+\varphi ^{n-2},\\\psi ^{n}&=\psi ^{n-1}+\psi ^{n-2}.\end{aligned}}}
11415:
10367:
7818:{\displaystyle {\begin{pmatrix}1&1\\1&0\end{pmatrix}}^{n}={\begin{pmatrix}F_{n+1}&F_{n}\\F_{n}&F_{n-1}\end{pmatrix}}.}
28124:
25122:
24929:
24554:
24326:
24028:
24003:
23433:
15175:
11167:
99:
Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the
20287:
15117:
11276:
9774:
2474:
1861:
28555:
26183:
24416:
19902:
19514:
5411:
4401:
23541:
are: . In this way Indian prosodists were led to discover the Fibonacci sequence, as we have observed in Section 1.2.8 (from v.1)
14320:
10364:
A similar argument, grouping the sums by the position of the first 1 rather than the first 2 gives two more identities:
28840:
25537:
22342:. This means that every positive integer can be written as a sum of Fibonacci numbers, where any one number is used once at most.
21964:
10931:
26113:- animation of sequence, spiral, golden ratio, rabbit pair growth. Examples in art, music, architecture, nature, and astronomy
22349:
distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers. This is known as
13728:{\displaystyle F_{4n}=4F_{n}F_{n+1}\left({F_{n+1}}^{2}+2{F_{n}}^{2}\right)-3{F_{n}}^{2}\left({F_{n}}^{2}+2{F_{n+1}}^{2}\right)}
3882:
1573:{\displaystyle F_{n}={\frac {\varphi ^{n}-(-\varphi )^{-n}}{\sqrt {5}}}={\frac {\varphi ^{n}-(-\varphi )^{-n}}{2\varphi -1}}.}
28565:
26089:
26031:
26013:
25985:
25963:
25944:
25704:
25625:
25477:
25315:
25275:
25196:
24956:
24638:
24592:
24183:
24052:
23954:
23918:
23887:
23615:
23527:
23464:
25743:
Yanega, D. 1996. Sex ratio and sex allocation in sweat bees (Hymenoptera: Halictidae). J. Kans. Ent. Soc. 69 Suppl.: 98-115.
209:
84:
poetry formed from syllables of two lengths. They are named after the Italian mathematician Leonardo of Pisa, also known as
28866:
28861:
28856:
23353:
21197:
9291:
265:
25428:
3395:
3278:
28785:
26990:
26176:
961:
24834:
Broughan, Kevin A.; GonzĂĄlez, Marcos J.; Lewis, Ryan H.; Luca, Florian; MejĂa Huguet, V. Janitzio; TogbĂ©, Alain (2011),
1020:-th month, the number of pairs of rabbits is equal to the number of mature pairs (that is, the number of pairs in month
34:
in which each number is the sum of the two preceding ones. Numbers that are part of the Fibonacci sequence are known as
28729:
28560:
28320:
28267:
26985:
24209:
20841:{\displaystyle 5{F_{14}}^{2}=710645\equiv 0{\pmod {29}}\;\;{\text{ and }}\;\;5{F_{15}}^{2}=1860500\equiv 5{\pmod {29}}}
18104:
9958:
9009:
2752:
22779:
advanced the idea that real instances can in part be understood as the expression of certain algebraic constraints on
28709:
27000:
26059:
25586:
24382:
23812:
23243:, when a beam of light shines at an angle through two stacked transparent plates of different materials of different
11041:
5871:{\displaystyle {\vec {F}}_{0}={1 \choose 0}={\frac {1}{\sqrt {5}}}{\vec {\mu }}-{\frac {1}{\sqrt {5}}}{\vec {\nu }},}
26980:
10291:
28719:
27693:
27273:
25617:
23855:
23828:
3237:
1223:{\displaystyle F_{n}={\frac {\varphi ^{n}-\psi ^{n}}{\varphi -\psi }}={\frac {\varphi ^{n}-\psi ^{n}}{\sqrt {5}}},}
163:
and the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as
17935:
Fibonacci primes with thousands of digits have been found, but it is not known whether there are infinitely many.
3950:
as a linear function of lower powers, which in turn can be decomposed all the way down to a linear combination of
3864:
3098:{\displaystyle n_{\mathrm {largest} }(F)=\left\lfloor \log _{\varphi }{\sqrt {5}}(F+1/2)\right\rfloor ,\ F\geq 0,}
917:(c. 100 BCâc. 350 AD). However, the clearest exposition of the sequence arises in the work of
28611:
28601:
28144:
22602:
16110:{\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k+1}}{\sum _{j=1}^{k}{F_{j}}^{2}}}={\frac {{\sqrt {5}}-1}{2}}.}
10925:
10921:
10875:
7611:
24606:
21292:
use the Fibonacci recursion with other starting points to generate sequences in which all numbers are composite.
11925:
986:
where it is used to calculate the growth of rabbit populations. Fibonacci considers the growth of an idealized (
26995:
25265:
24485:
22145:
sâthey are 0000, 0001, 0010, 0100, 0101, 1000, 1001, and 1010. Such strings are the binary representations of
20459:{\displaystyle 5{F_{6}}^{2}=320\equiv -5{\pmod {13}}\;\;{\text{ and }}\;\;5{F_{7}}^{2}=845\equiv 0{\pmod {13}}}
15012:
12584:
5327:{\displaystyle {F_{k+2} \choose F_{k+1}}={\begin{pmatrix}1&1\\1&0\end{pmatrix}}{F_{k+1} \choose F_{k}}}
20071:{\displaystyle 5{F_{5}}^{2}=125\equiv 4{\pmod {11}}\;\;{\text{ and }}\;\;5{F_{6}}^{2}=320\equiv 1{\pmod {11}}}
15240:
3345:
1393:{\displaystyle \psi ={\frac {1-{\sqrt {5}}}{2}}=1-\varphi =-{1 \over \varphi }\approx -0.61803\,39887\ldots .}
28724:
28626:
28260:
28194:
28117:
27779:
26153:
24903:
22625:
16476:
11102:
10785:
8929:
4738:
20:
25389:
23776:
19689:{\displaystyle 5{F_{3}}^{2}=20\equiv -1{\pmod {7}}\;\;{\text{ and }}\;\;5{F_{4}}^{2}=45\equiv -4{\pmod {7}}}
15501:{\displaystyle s(0.001)={\frac {0.001}{0.998999}}={\frac {1000}{998999}}=0.001001002003005008013021\ldots .}
12534:
9875:
4486:
1632:
28752:
27445:
27095:
26764:
26557:
24580:
23517:
23302:
being shifted. To convert from kilometers to miles, shift the register down the Fibonacci sequence instead.
15412:
equal to any of 0.01, 0.001, 0.0001, etc. lays out the first Fibonacci numbers in the decimal expansion of
11035:
8518:
3708:
1107:
422:
117:
24445:
16231:{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{F_{2k}}}={\sqrt {5}}\left(L(\psi ^{2})-L(\psi ^{4})\right)}
1405:
28734:
27621:
27480:
27311:
27125:
27115:
26769:
26749:
26148:
24341:
André-Jeannin, Richard (1989), "Irrationalité de la somme des inverses de certaines suites récurrentes",
10998:
10739:
868:("the two are mixed") and scholars who interpret it in context as saying that the number of patterns for
27450:
25804:
22776:
17902:
14271:
28906:
28901:
28616:
27570:
27193:
27035:
26950:
26759:
26741:
26635:
26625:
26615:
26451:
26143:
26131:
26110:
25936:
23522:, vol. 4. Generating All Trees â History of Combinatorial Generation, AddisonâWesley, p. 50,
22818:
22639:
22332:
19002:
13740:
8867:
7843:
4362:
4325:
106:
27475:
25768:
15107:{\displaystyle s(z)={\frac {1}{\sqrt {5}}}\left({\frac {1}{1-\varphi z}}-{\frac {1}{1-\psi z}}\right)}
8808:
6677:{\displaystyle {\begin{aligned}A&=S\Lambda S^{-1},\\A^{n}&=S\Lambda ^{n}S^{-1},\end{aligned}}}
28871:
28606:
28596:
28586:
27698:
27243:
26864:
26650:
26645:
26640:
26630:
26607:
26003:
25528:
23364:
23347:
23305:
The measured values of voltages and currents in the infinite resistor chain circuit (also called the
23025:. This is under the unrealistic assumption that the ancestors at each level are otherwise unrelated.
22329:
22100:
9680:
3558:
27455:
25330:
Adelson-Velsky, Georgy; Landis, Evgenii (1962), "An algorithm for the organization of information",
25117:"Sequence A001175 (Pisano periods (or Pisano numbers): period of Fibonacci numbers mod n)"
22584:
of two integers: the worst case input for this algorithm is a pair of consecutive Fibonacci numbers.
19065:
18177:
17363:
9872:
based on the location of the first 2. Specifically, each set consists of those sequences that start
8755:
921:(c. 700 AD), whose own work is lost, but is available in a quotation by Gopala (c. 1135):
373:
28110:
27120:
27030:
26683:
26082:
Fibonacci's Liber Abaci: A Translation into Modern English of Leonardo Pisano's Book of Calculation
25678:
24377:, Dolciani Mathematical Expositions, vol. 9, American Mathematical Society, pp. 135â136,
23428:"Sequence A000045 (Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1)"
23313:
Brasch et al. 2012 show how a generalized Fibonacci sequence also can be connected to the field of
21521:{\displaystyle F_{n}=\sum _{k=0}^{\left\lfloor {\frac {n-1}{2}}\right\rfloor }{\binom {n-k-1}{k}}.}
15231:
11496:
8584:
can be interpreted as the number of (possibly empty) sequences of 1s and 2s whose sum is
1303:
25138:
18069:) every Fibonacci number has a prime factor that is not a factor of any smaller Fibonacci number (
12502:
5751:{\displaystyle {\vec {\mu }}={\varphi \choose 1},\quad {\vec {\nu }}={-\varphi ^{-1} \choose 1}.}
4776:
3926:
28778:
28701:
28523:
27809:
27774:
27560:
27470:
27344:
27319:
27228:
27218:
26940:
26830:
26812:
26732:
24090:
23742:
23028:
22758:
22650:
22581:
22350:
18070:
18024:
18013:
17039:
11998:
25859:
Brasch, T. von; Byström, J.; Lystad, L.P. (2012), "Optimal Control and the Fibonacci Sequence",
25696:
25690:
24924:"Sequence A235383 (Fibonacci numbers that are the product of other Fibonacci numbers)"
1060:: In a growing idealized population, the number of rabbit pairs form the Fibonacci sequence. At
28363:
28310:
28242:
28069:
27339:
27213:
26844:
26620:
26400:
26327:
26106:
Fibonacci Sequence and Golden Ratio: Mathematics in the Modern World - Mathuklasan with Sir Ram
24199:
23781:
22676:
22573:
22541:
21986:
17898:
15516:
13081:{\displaystyle F_{2n}={F_{n+1}}^{2}-{F_{n-1}}^{2}=F_{n}\left(F_{n+1}+F_{n-1}\right)=F_{n}L_{n}}
11158:
7974:
6562:
6351:
4912:{\displaystyle \varphi ^{n}={\frac {F_{n}{\sqrt {5}}\pm {\sqrt {5{F_{n}}^{2}+4(-1)^{n}}}}{2}}.}
4048:
2360:{\displaystyle \left\{{\begin{aligned}a+b&=0\\\varphi a+\psi b&=1\end{aligned}}\right.}
1099:
125:
24370:
23925:
23605:
23454:
23045:, which he received from his father. The male counts as the "origin" of his own X chromosome (
22318:
sâthey are 0001, 0111, 0101, 1000, 1010, 1110. There is an equivalent statement about subsets.
10529:
4605:
4569:
3953:
3842:
28570:
28315:
28184:
28033:
27673:
27324:
27178:
27105:
26260:
25290:
23272:
23194:
23157:
23120:
23083:
23048:
22097:
21333:
17085:
17052:
16922:
16880:
14985:
14388:
11240:
10258:
9647:
9614:
8706:
8673:
8640:
7850:
340:
17997:
Fibonacci number is 144. Attila PethĆ proved in 2001 that there is only a finite number of
10577:
2854:{\displaystyle F_{n}=\left\lfloor {\frac {\varphi ^{n}}{\sqrt {5}}}\right\rceil ,\ n\geq 0.}
28681:
28518:
28287:
27966:
27860:
27824:
27565:
27288:
27268:
27085:
26754:
26542:
25826:
25285:
25162:
24889:
24851:
24813:
24725:
24674:
24602:
24518:
24356:
23861:
23833:
23674:
22754:
22687:
22597:
22325:
21745:
21412:
21408:
21400:
21295:
Letting a number be a linear function (other than the sum) of the 2 preceding numbers. The
21169:
14958:
14236:{\displaystyle s(z)=\sum _{k=0}^{\infty }F_{k}z^{k}=0+z+z^{2}+2z^{3}+3z^{4}+5z^{5}+\dots .}
10700:
10571:
10523:
9732:
9356:
8613:
8560:
5483:
3681:
3654:
3227:{\displaystyle \log _{\varphi }(x)=\ln(x)/\ln(\varphi )=\log _{10}(x)/\log _{10}(\varphi )}
1593:
1057:
996:
27045:
26514:
25576:
23672:
Singh, Parmanand (1985), "The So-called Fibonacci Numbers in Ancient and Medieval India",
23482:
Singh, Parmanand (1985), "The So-called Fibonacci numbers in ancient and medieval India",
15523:. For example, the sum of every odd-indexed reciprocal Fibonacci number can be written as
15415:
15297:
9759:
In a similar manner it may be shown that the sum of the first Fibonacci numbers up to the
1006:
At the end of the second month they produce a new pair, so there are 2 pairs in the field.
8:
28877:
28817:
28661:
28528:
27688:
27552:
27547:
27515:
27278:
27253:
27248:
27223:
27153:
27149:
27080:
26970:
26802:
26598:
26567:
25682:
25494:
24317:"Sequence A079586 (Decimal expansion of Sum_{k>=1} 1/F(k) where F(k) is the
23283:
22577:
21217:
Some specific examples that are close, in some sense, to the Fibonacci sequence include:
21207:
21203:
16858:{\displaystyle \sum _{k=0}^{N}{\frac {1}{F_{2^{k}}}}=3-{\frac {F_{2^{N}-1}}{F_{2^{N}}}}.}
15324:
14080:
11390:
11008:
9849:
9588:
9562:
8587:
8554:
5183:
3971:
3552:
3339:
204:
172:
101:
25830:
24729:
24422:
24144:
21772:
To see how the formula is used, we can arrange the sums by the number of terms present:
21202:
The Fibonacci sequence is one of the simplest and earliest known sequences defined by a
11749:{\displaystyle \sum _{i=1}^{n}F_{i}^{2}+{F_{n+1}}^{2}=F_{n+1}\left(F_{n}+F_{n+1}\right)}
199:
connecting the opposite corners of squares in the Fibonacci tiling (see preceding image)
28771:
28591:
28502:
28487:
28459:
28439:
28091:
27845:
27840:
27754:
27728:
27626:
27605:
27377:
27258:
27208:
27130:
27100:
27040:
26807:
26787:
26718:
26069:
25886:
25842:
25359:
Avriel, M; Wilde, DJ (1966), "Optimality of the Symmetric Fibonacci Search Technique",
25342:
25247:
25095:
24817:
24741:
24715:
24279:
24271:
24140:
23734:
23396:
23330:
23326:
23292:
23276:
22906:
22745:
pointed out the presence of the Fibonacci sequence in nature, using it to explain the (
22699:
22530:
22369:
21399:
The Fibonacci numbers are the sums of the diagonals (shown in red) of a left-justified
21289:
21173:
17288:
16736:{\displaystyle \sum _{k=0}^{\infty }{\frac {1}{F_{2^{k}}}}={\frac {7-{\sqrt {5}}}{2}},}
15395:
14251:
14051:{\displaystyle F_{kn+c}=\sum _{i=0}^{k}{k \choose i}F_{c+i}{F_{n}}^{i}{F_{n-1}}^{k-i}.}
13896:{\displaystyle F_{kn+c}=\sum _{i=0}^{k}{k \choose i}F_{c-i}{F_{n}}^{i}{F_{n+1}}^{k-i}.}
12575:
Numerous other identities can be derived using various methods. Here are some of them:
10980:
10062:
Following the same logic as before, by summing the cardinality of each set we see that
7471:
6572:
4770:
4044:
830:
73:
26975:
26116:
24082:
9380:
sequences into two non-overlapping sets where all sequences either begin with 1 or 2:
5635:{\displaystyle \psi =-\varphi ^{-1}={\tfrac {1}{2}}{\bigl (}1-{\sqrt {5}}~\!{\bigr )}}
5045:{\displaystyle \varphi ^{n}=F_{n}\varphi +F_{n-1}=(F_{n}{\sqrt {5}}+F_{n}+2F_{n-1})/2}
2958:{\displaystyle n(F)=\left\lfloor \log _{\varphi }{\sqrt {5}}F\right\rceil ,\ F\geq 1.}
28691:
28492:
28464:
28418:
28408:
28388:
28373:
28218:
28199:
28159:
28087:
27985:
27930:
27784:
27759:
27733:
27188:
27183:
27110:
27090:
27075:
26797:
26779:
26698:
26688:
26673:
26436:
26085:
26084:, Sources and Studies in the History of Mathematics and Physical Sciences, Springer,
26055:
26027:
26009:
25981:
25959:
25940:
25700:
25653:
25621:
25582:
25473:
25311:
25271:
25192:
24952:
24877:
24821:
24801:
24634:
24588:
24378:
24283:
24234:
24230:
24205:
24179:
24048:
23998:"Sequence A002390 (Decimal expansion of natural logarithm of golden ratio)"
23950:
23914:
23883:
23808:
23692:
23611:
23523:
23496:
23460:
23296:
22646:
22339:
22321:
21146:
18005:
17970:
16640:
15878:
If we add 1 to each Fibonacci number in the first sum, there is also the closed form
13736:
11097:
11002:
8746:
7464:
4801:
1111:
27510:
25846:
24745:
15961:{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{1+F_{2k-1}}}={\frac {\sqrt {5}}{2}},}
13267:{\displaystyle F_{3n}=2{F_{n}}^{3}+3F_{n}F_{n+1}F_{n-1}=5{F_{n}}^{3}+3(-1)^{n}F_{n}}
28676:
28497:
28423:
28413:
28393:
28295:
28174:
28021:
27814:
27400:
27372:
27362:
27354:
27238:
27203:
27198:
27165:
26859:
26822:
26713:
26708:
26703:
26693:
26665:
26552:
26499:
26456:
26395:
25995:
25973:
25890:
25876:
25868:
25834:
25649:
25443:
25310:, vol. 1: Fundamental Algorithms (3rd ed.), AddisonâWesley, p. 343,
25087:
24793:
24733:
24662:
24504:
24494:
24263:
23942:
23687:
23491:
23244:
22714:
22682:
22526:
22522:
22354:
22146:
21340:
17974:
17696:
17046:
7601:{\displaystyle \varphi =1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+\ddots }}}}}}.}
1115:
834:
789:
31:
26504:
25758:"Growing the Family Tree: The Power of DNA in Reconstructing Family Relationships"
25078:
Freyd, Peter; Brown, Kevin S. (1993), "Problems and Solutions: Solutions: E3410",
23189:. Five great-great-grandparents contributed to the male descendant's X chromosome
22705:
22562:
21168:, which includes as a subproblem a special instance of the problem of finding the
18091:) are the only Fibonacci numbers that are the product of other Fibonacci numbers.
1046:
28454:
28383:
28169:
27997:
27886:
27819:
27745:
27668:
27642:
27460:
27173:
26965:
26935:
26925:
26920:
26586:
26494:
26441:
26285:
26225:
26045:
25281:
25212:
25158:
24944:
24885:
24847:
24809:
24687:
PethĆ, Attila (2001), "Diophantine properties of linear recursive sequences II",
24670:
24598:
24514:
24397:
24352:
23908:
23334:
23306:
23269:, there are three reflection paths, not two, one for each of the three surfaces.)
23037:
It has similarly been noticed that the number of possible ancestors on the human
22762:
22671:
Some Agile teams use a modified series called the "Modified Fibonacci Series" in
21185:
18099:
17977:, there are therefore also arbitrarily long runs of composite Fibonacci numbers.
17914:
17547:
3548:
1045:
The name "Fibonacci sequence" was first used by the 19th-century number theorist
25765:
Proceedings of the First Symposium on Bioinformatics and Biotechnology (BIOT-04)
24835:
24737:
23946:
22773:) of plants were frequently expressed as fractions involving Fibonacci numbers.
2439:{\displaystyle a={\frac {1}{\varphi -\psi }}={\frac {1}{\sqrt {5}}},\quad b=-a,}
1285:{\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}\approx 1.61803\,39887\ldots }
807:) ways of arranging long and short syllables in a cadence of length six. Eight (
68:
whose side lengths are successive Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13 and 21
28686:
28671:
28666:
28345:
28330:
28179:
28002:
27870:
27855:
27719:
27683:
27658:
27534:
27505:
27490:
27367:
27263:
27233:
26960:
26915:
26792:
26390:
26385:
26380:
26352:
26337:
26250:
26235:
26213:
26200:
25686:
25451:
24666:
24569:
23875:
23683:
23358:
23227:
22721:
Fibonacci sequences appear in biological settings, such as branching in trees,
22672:
22632:
22518:
22365:
18020:
17617:
17260:{\displaystyle \gcd(F_{n},F_{n+1})=\gcd(F_{n},F_{n+2})=\gcd(F_{n+1},F_{n+2})=1}
16925:. In fact, the Fibonacci sequence satisfies the stronger divisibility property
16324:{\displaystyle \textstyle L(q):=\sum _{k=1}^{\infty }{\frac {q^{k}}{1-q^{k}}},}
16239:
15520:
14246:
10245:{\displaystyle F_{n+2}=F_{n}+F_{n-1}+...+|\{(1,1,...,1,2)\}|+|\{(1,1,...,1)\}|}
7978:
4563:
2968:
176:
121:
113:
110:
88:, who introduced the sequence to Western European mathematics in his 1202 book
26104:
26005:
Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity
25872:
25719:
25405:
24759:
24509:
23078:), and at his parents' generation, his X chromosome came from a single parent
22345:
Moreover, every positive integer can be written in a unique way as the sum of
22026:
ways one can climb a staircase of 5 steps, taking one or two steps at a time:
28895:
28651:
28325:
28189:
28133:
27925:
27909:
27850:
27804:
27500:
27485:
27395:
26678:
26547:
26509:
26466:
26347:
26332:
26322:
26280:
26270:
26245:
26168:
25999:
25838:
25509:
24881:
24805:
24295:
23399:
twenty-one. In this way, the process should be followed in all mÄtrÄ-váčttas"
23370:
22545:
22244:
sâthey are 0000, 0011, 0110, 1100, 1111. Equivalently, the number of subsets
21980:
21344:
21222:
21161:
21136:
17998:
17994:
17680:
15291:
13744:
10736:-th Fibonacci numbers. To see this, begin with a Fibonacci rectangle of size
7965:
4639:
944:
913:
794:
192:
26127:
24585:
Connections in Discrete Mathematics: A Celebration of the Work of Ron Graham
21947:
21395:
124:
used for interconnecting parallel and distributed systems. They also appear
105:. Applications of Fibonacci numbers include computer algorithms such as the
28656:
28398:
28340:
28208:
28154:
28149:
27961:
27950:
27865:
27703:
27678:
27595:
27495:
27465:
27440:
27424:
27329:
27296:
27019:
26930:
26869:
26446:
26342:
26275:
26255:
26230:
26122:
26051:
25614:
Lindenmayer Systems, Fractals, and Plants (Lecture Notes in Biomathematics)
25447:
25303:
24534:, Richard Crandall, Carl Pomerance, Springer, second edition, 2005, p. 142.
24499:
23601:
23513:
23042:
23038:
22971:
22914:
22746:
21236:
21177:
21165:
17924:
17278:
15975:
14084:
13104:
1295:
1080:
908:
196:
168:
148:
23367: â Randomized mathematical sequence based upon the Fibonacci sequence
23323:
included the Fibonacci sequence in some of his artworks beginning in 1970.
21922:{\displaystyle \textstyle {\binom {5}{0}}+{\binom {4}{1}}+{\binom {3}{2}}}
21328:. If the coefficient of the preceding value is assigned a variable value
7981:, and one may easily deduce the second one from the first one by changing
7444:{\displaystyle F_{n}={\cfrac {\varphi ^{n}-(-\varphi )^{-n}}{\sqrt {5}}}.}
5550:{\displaystyle \varphi ={\tfrac {1}{2}}{\bigl (}1+{\sqrt {5}}~\!{\bigr )}}
2881:. This formula is easily inverted to find an index of a Fibonacci number
2720:{\textstyle \left|{\frac {\psi ^{n}}{\sqrt {5}}}\right|<{\frac {1}{2}}}
28809:
28403:
28350:
28213:
27920:
27795:
27600:
27064:
26955:
26910:
26905:
26655:
26562:
26461:
26290:
26265:
26240:
26041:
25267:
Introduction to Circle Packing: The Theory of Discrete Analytic Functions
25112:
24919:
24544:
24312:
24018:
23993:
23423:
22979:
22770:
22722:
22614:
22610:
21296:
21229:
18992:{\displaystyle F_{p-\left({\frac {p}{5}}\right)}\equiv 0{\pmod {p^{2}}}.}
17027:{\displaystyle \gcd(F_{a},F_{b},F_{c},\ldots )=F_{\gcd(a,b,c,\ldots )}\,}
14067:
in this formula, one gets again the formulas of the end of above section
10697:
or in words, the sum of the squares of the first Fibonacci numbers up to
9559:
Excluding the first element, the remaining terms in each sequence sum to
9552:{\displaystyle F_{n}=|\{(1,...),(1,...),...\}|+|\{(2,...),(2,...),...\}|}
8742:
8537:
7977:
can be derived (they are obtained from two different coefficients of the
7460:
5643:
3975:
3832:{\displaystyle \lim _{n\to \infty }{\frac {F_{n+m}}{F_{n}}}=\varphi ^{m}}
1003:
At the end of the first month, they mate, but there is still only 1 pair.
973:
955:
933:
129:
91:
25881:
25251:
22506:{\displaystyle (F_{n}F_{n+3})^{2}+(2F_{n+1}F_{n+2})^{2}={F_{2n+3}}^{2}.}
1582:
To see the relation between the sequence and these constants, note that
28252:
28057:
28038:
27334:
26945:
26047:
The Golden Ratio: The Story of Phi, the World's Most Astonishing Number
25099:
24964:
24797:
24275:
23941:, Einblick in die Wissenschaft, Vieweg+Teubner Verlag, pp. 87â98,
23373: â Infinite matrix of integers derived from the Fibonacci sequence
23320:
22888:{\displaystyle \theta ={\frac {2\pi }{\varphi ^{2}}}n,\ r=c{\sqrt {n}}}
22780:
22766:
22661:
22588:
22537:
22361:
18009:
5479:
2971:
gives the largest index of a Fibonacci number that is not greater than
25640:
Vogel, Helmut (1979), "A better way to construct the sunflower head",
24865:
24023:"Sequence A097348 (Decimal expansion of arccsch(2)/log(10))"
22096:
The Fibonacci numbers can be found in different ways among the set of
13522:{\displaystyle F_{3n+2}={F_{n+1}}^{3}+3{F_{n+1}}^{2}F_{n}+{F_{n}}^{3}}
13396:{\displaystyle F_{3n+1}={F_{n+1}}^{3}+3F_{n+1}{F_{n}}^{2}-{F_{n}}^{3}}
10570:-th Fibonacci number, and the sum of the first Fibonacci numbers with
4728:{\displaystyle F_{n}=(\varphi ^{n}-(-1)^{n}\varphi ^{-n})/{\sqrt {5}}}
3839:, because the ratios between consecutive Fibonacci numbers approaches
3642:{\displaystyle \lim _{n\to \infty }{\frac {F_{n+1}}{F_{n}}}=\varphi .}
864: 450 BCâ200 BC). Singh cites Pingala's cryptic formula
28794:
28335:
27663:
27590:
27582:
27387:
27301:
26419:
26123:
Scientists find clues to the formation of Fibonacci spirals in nature
25933:
Strange Curves, Counting Rabbits, and Other Mathematical Explorations
24720:
24239:
23937:
Beutelspacher, Albrecht; Petri, Bernhard (1996), "Fibonacci-Zahlen",
23314:
22809:
22734:
22730:
22726:
22710:
22197:
22090:
20876:(as the products of odd prime divisors) are congruent to 1 modulo 4.
17543:
9278:{\displaystyle F_{5}=5=|\{(1,1,1,1),(1,1,2),(1,2,1),(2,1,1),(2,2)\}|}
8517:
These last two identities provide a way to compute Fibonacci numbers
983:
949:
918:
141:
137:
133:
85:
25091:
24976:
24267:
22360:
Starting with 5, every second Fibonacci number is the length of the
17045:
In particular, any three consecutive Fibonacci numbers are pairwise
10861:
3534:{\displaystyle n\log _{b}\varphi ={\frac {n\log \varphi }{\log b}}.}
1052:
60:
28283:
27764:
24174:
Vorobiev, NikolaÄ Nikolaevich; Martin, Mircea (2002), "Chapter 1",
22784:
22750:
22738:
22618:
22557:
22299:. For example, out of the 16 binary strings of length 4, there are
22230:. For example, out of the 16 binary strings of length 4, there are
22131:. For example, out of the 16 binary strings of length 4, there are
2786:
1091:
187:
81:
25007:
22665:
17284:
divides a Fibonacci number that can be determined by the value of
15008:
cancel out because of the defining Fibonacci recurrence relation.
28102:
27769:
27428:
27422:
26160:
25042:
24988:
22790:
21212:
homogeneous linear difference equation with constant coefficients
6350:
th element in the Fibonacci series may be read off directly as a
2746:
987:
857:
77:
22757:
most often have petals in counts of Fibonacci numbers. In 1830,
22675:, as an estimation tool. Planning Poker is a formal part of the
22324:
was able to show that the Fibonacci numbers can be defined by a
12799:{\displaystyle {F_{n}}^{2}-F_{n+r}F_{n-r}=(-1)^{n-r}{F_{r}}^{2}}
8553:
Most identities involving Fibonacci numbers can be proved using
24116:
23240:
22805:
22742:
22664:
the original audio wave similar to logarithmic methods such as
22368:
with integer sides, or in other words, the largest number in a
22104:
20865:
are congruent to 1 modulo 4, implying that all odd divisors of
15223:{\displaystyle \psi ={\tfrac {1}{2}}\left(1-{\sqrt {5}}\right)}
15165:{\textstyle \varphi ={\tfrac {1}{2}}\left(1+{\sqrt {5}}\right)}
8703:, meaning the empty sequence "adds up" to 0. In the following,
8532:
arithmetic operations. This matches the time for computing the
991:
964:
showing (in box on right) 13 entries of the Fibonacci sequence:
65:
26484:
24866:"On Perfect numbers which are ratios of two Fibonacci numbers"
22093:
until a single step, of which there is only one way to climb.
15974:
sum of squared Fibonacci numbers giving the reciprocal of the
11377:{\displaystyle \sum _{i=1}^{n}F_{i}+F_{n+1}=F_{n+1}+F_{n+2}-1}
10949:
5164:{\displaystyle 5{F_{n}}^{2}+4(-1)^{n}=(F_{n}+2F_{n-1})^{2}\,.}
2277:
must be the Fibonacci sequence. This is the same as requiring
856:
Knowledge of the Fibonacci sequence was expressed as early as
25238:
Pagni, David (September 2001), "Fibonacci Meets Pythagoras",
24784:
Luca, Florian (2000), "Perfect Fibonacci and Lucas numbers",
24117:"The Golden Ratio, Fibonacci Numbers and Continued Fractions"
23710:, Jodhpur: Rajasthan Oriental Research Institute, p. 101
23288:
22978:
If an egg is laid but not fertilized, it produces a male (or
22910:
22657:
22525:
with a Fibonacci number of nodes that has been proposed as a
22207:
is to replace 1 with 0 and 2 with 10, and drop the last zero.
21164:. Determining a general formula for the Pisano periods is an
7849:
Taking the determinant of both sides of this equation yields
5471:{\displaystyle {\vec {F}}_{n}=\mathbf {A} ^{n}{\vec {F}}_{0}}
3453:
28763:
22966:
Fibonacci numbers also appear in the ancestral pedigrees of
21141:
If the members of the Fibonacci sequence are taken mod
17969:, any Fibonacci prime must have a prime index. As there are
17672:{\displaystyle n\mid F_{n\;-\,\left({\frac {5}{n}}\right)},}
17602:{\displaystyle p\mid F_{p\;-\,\left({\frac {5}{p}}\right)}.}
16119:
The sum of all even-indexed reciprocal Fibonacci numbers is
8670:, meaning no such sequence exists whose sum is â1, and
5399:{\displaystyle {\vec {F}}_{k+1}=\mathbf {A} {\vec {F}}_{k},}
175:
and with the Fibonacci numbers form a complementary pair of
26163:
25116:
24923:
24548:
24316:
24022:
23997:
23427:
22653:
21221:
Generalizing the index to negative integers to produce the
19288:
18292:
17526:
16743:
which follows from the closed form for its partial sums as
10854:
4642:. This is because Binet's formula, which can be written as
2354:
26026:, Springer Monographs in Mathematics, New York: Springer,
25931:
Ball, Keith M (2003), "8: Fibonacci's Rabbits Revisited",
24442:
Williams, H. C. (1982), "A note on the Fibonacci quotient
18023:. More generally, no Fibonacci number other than 1 can be
16877:
Every third number of the sequence is even (a multiple of
10970:{\displaystyle \operatorname {Seq} ({\mathcal {Z+Z^{2}}})}
2863:
In fact, the rounding error quickly becomes very small as
26137:
25492:
Brousseau, A (1969), "Fibonacci Statistics in Conifers",
24833:
24228:
23757:
22967:
22587:
Fibonacci numbers are used in a polyphase version of the
17712:
15519:
Fibonacci numbers can sometimes be evaluated in terms of
12915:{\displaystyle F_{m}F_{n+1}-F_{m+1}F_{n}=(-1)^{n}F_{m-n}}
11842:{\displaystyle \sum _{i=1}^{n+1}F_{i}^{2}=F_{n+1}F_{n+2}}
7660:-st convergent can be found from the recurrence relation
6829:
th element in the Fibonacci series is therefore given by
4558:
Binet's formula provides a proof that a positive integer
3651:
This convergence holds regardless of the starting values
3551:
observed that the ratio of consecutive Fibonacci numbers
167:
increases. Fibonacci numbers are also closely related to
25677:
24689:
Acta Mathematica Academiae Paedagogicae NyĂregyhĂĄziensis
23882:, The Mathematical Association of America, p. 153,
23829:"The Fibonacci Numbers and Golden section in Nature â 1"
23714:
23570:
22985:
If, however, an egg is fertilized, it produces a female.
21528:
This can be proved by expanding the generating function
18027:, and no ratio of two Fibonacci numbers can be perfect.
11912:{\displaystyle {\sqrt {5}}F_{n}=\varphi ^{n}-\psi ^{n}.}
25956:
The Art of Proof: Basic Training for Deeper Mathematics
25556:
23974:
23633:
23631:
23546:
21979:
ways to do this (equivalently, it's also the number of
15664:
and the sum of squared reciprocal Fibonacci numbers as
7566:
7554:
7540:
7528:
7514:
7502:
7423:
7379:
6561:
Equivalently, the same computation may be performed by
6522:
6500:
6477:
6465:
6434:
6412:
6389:
6377:
6266:
6244:
6221:
6209:
6155:
6133:
6110:
6098:
5174:
In particular, the left-hand side is a perfect square.
3923:
this expression can be used to decompose higher powers
2469:
to be arbitrary constants, a more general solution is:
25784:
25111:
24918:
24653:
Cohn, J. H. E. (1964), "On square Fibonacci numbers",
24549:"Sequence A005478 (Prime Fibonacci numbers)"
24543:
24311:
24017:
23992:
23508:
23506:
23422:
23350: â Organization for research on Fibonacci numbers
22905:
is a constant scaling factor; the florets thus lie on
21845:
20625:
20598:
20558:
20531:
20499:
20240:
20213:
20170:
20143:
20111:
19855:
19828:
19788:
19761:
19729:
19464:
19437:
19394:
19367:
19335:
19205:
19178:
19096:
19069:
18816:
18720:
18652:
18556:
18460:
18116:
17825:
17745:
16338:
16248:
15243:
15186:
15128:
15120:
14391:
12537:
12505:
11928:
11157:
Fibonacci identities often can be easily proved using
10522:
In words, the sum of the first Fibonacci numbers with
9846:
This may be seen by dividing all sequences summing to
7741:
7696:
7569:
7557:
7543:
7531:
7517:
7505:
7426:
7382:
7271:
7197:
7148:
7027:
6766:
6706:
6525:
6503:
6480:
6468:
6437:
6415:
6392:
6380:
6269:
6247:
6224:
6212:
6158:
6136:
6113:
6101:
5590:
5505:
5251:
2676:
1596:
28632:
1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⯠(inverses of primes)
28622:
1 â 1 + 2 â 6 + 24 â 120 + ⯠(alternating factorials)
27148:
24705:
24448:
23913:(2nd, illustrated ed.). CRC Press. p. 260.
23906:
23477:
23475:
23197:
23160:
23123:
23086:
23051:
22831:
22378:
21989:
21844:
21748:
21534:
21421:
20885:
20728:
20675:
20490:
20343:
20290:
20102:
19958:
19905:
19720:
19570:
19517:
19326:
19021:
18925:
18447:
18318:
18150:
18107:
17739:
17626:
17556:
17357:
17124:
17088:
17055:
16931:
16883:
16753:
16659:
16485:
16337:
16247:
16125:
15984:
15884:
15670:
15529:
15447:
15418:
15398:
15383:{\displaystyle s(z)=s\!\left(-{\frac {1}{z}}\right).}
15334:
15300:
15178:
15021:
14988:
14961:
14436:
14323:
14274:
14254:
14094:
13913:
13758:
13534:
13408:
13282:
13117:
12927:
12818:
12694:
12681:{\displaystyle {F_{n}}^{2}-F_{n+1}F_{n-1}=(-1)^{n-1}}
12595:
12578:
12025:
12001:
11862:
11761:
11623:
11610:{\displaystyle \sum _{i=1}^{n}F_{i}^{2}=F_{n}F_{n+1}}
11541:
11499:
11418:
11393:
11279:
11243:
11170:
11105:
11044:
11011:
10983:
10934:
10878:
10788:
10742:
10703:
10690:{\displaystyle \sum _{i=1}^{n}F_{i}^{2}=F_{n}F_{n+1}}
10621:
10580:
10532:
10449:
10370:
10294:
10261:
10071:
9961:
9878:
9852:
9777:
9735:
9683:
9650:
9617:
9591:
9565:
9386:
9359:
9294:
9122:
9012:
8932:
8870:
8811:
8758:
8709:
8676:
8643:
8616:
8590:
8563:
8215:
7998:
7859:
7689:
7486:
7362:
6837:
6694:
6583:
6360:
5890:
5766:
5651:
5563:
5497:
5414:
5340:
5194:
5061:
4927:
4812:
4779:
4741:
4648:
4608:
4572:
4489:
4404:
4365:
4328:
4064:
3984:
3956:
3929:
3885:
3845:
3769:
3711:
3684:
3657:
3583:
3561:
3476:
3398:
3348:
3281:
3240:
3111:
2981:
2891:
2795:
2755:
2540:
2477:
2377:
2293:
1927:
1864:
1713:
1635:
1449:
1408:
1314:
1240:
1126:
425:
376:
343:
268:
212:
27533:
25908:
25896:
25822:
IEEE Transactions on Instrumentation and Measurement
25659:
25544:
25429:"Phyllotaxis as a Dynamical Self Organizing Process"
25329:
23628:
7942:{\displaystyle (-1)^{n}=F_{n+1}F_{n-1}-{F_{n}}^{2}.}
3431:{\displaystyle n\log _{10}\varphi \approx 0.2090\,n}
3316:{\displaystyle \log _{10}(\varphi )=0.208987\ldots }
144:'s bracts, though they do not occur in all species.
25858:
23962:
23503:
21184:, the Pisano period may be found as an instance of
20716:{\displaystyle F_{14}=377{\text{ and }}F_{15}=610.}
17679:where the Legendre symbol has been replaced by the
4036:{\displaystyle \varphi ^{n}=F_{n}\varphi +F_{n-1}.}
1027:) plus the number of pairs alive last month (month
147:Fibonacci numbers are also strongly related to the
25852:
25611:
25168:
24629:Honsberger, Ross (1985), "Mathematical Gems III",
24475:
24344:Comptes Rendus de l'Académie des Sciences, Série I
24155:
23936:
23805:Divine Proportion: Phi In Art, Nature, and Science
23582:
23472:
23216:
23179:
23142:
23105:
23070:
22887:
22638:A one-dimensional optimization method, called the
22505:
22001:
21921:
21761:
21734:
21520:
21339:Not adding the immediately preceding numbers. The
21096:
20840:
20715:
20662:
20458:
20330:
20277:
20070:
19945:
19892:
19688:
19557:
19504:
19294:
18991:
18903:
18431:
18298:
18136:
17883:
17671:
17601:
17532:
17259:
17107:
17074:
17026:
16902:
16857:
16735:
16629:
16465:
16323:
16230:
16109:
15960:
15868:
15654:
15500:
15433:
15404:
15382:
15315:
15282:
15222:
15164:
15106:
15000:
14974:
14945:
14422:
14375:
14307:
14260:
14235:
14050:
13895:
13727:
13521:
13395:
13266:
13080:
12914:
12798:
12680:
12559:
12523:
12491:
12011:
11987:
11911:
11841:
11748:
11609:
11527:
11483:
11405:
11376:
11262:
11229:
11142:
11088:
11023:
10989:
10969:
10912:
10845:
10774:
10716:
10689:
10596:
10554:
10514:
10435:
10353:
10280:
10244:
10051:
9947:
9864:
9836:
9748:
9721:
9669:
9636:
9603:
9577:
9551:
9372:
9345:
9277:
9107:
8997:
8917:
8855:
8796:
8733:
8695:
8662:
8629:
8602:
8576:
8507:
8189:
7941:
7817:
7600:
7443:
7346:
6815:
6676:
6551:
6336:
5870:
5750:
5634:
5549:
5470:
5398:
5326:
5163:
5044:
4911:
4792:
4761:
4727:
4630:
4594:
4543:
4455:
4390:
4351:
4314:
4035:
3962:
3942:
3913:
3851:
3831:
3748:
3697:
3670:
3641:
3570:
3533:
3430:
3373:
3315:
3267:
3226:
3097:
2957:
2853:
2777:
2719:
2655:
2522:
2438:
2359:
2228:
1909:
1838:
1689:
1621:
1572:
1433:
1392:
1284:
1222:
476:
411:
362:
319:
254:
56:0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ....
26532:
26074:(in French), vol. 1, Paris: Gauthier-Villars
26050:(First trade paperback ed.), New York City:
24864:Luca, Florian; MejĂa Huguet, V. Janitzio (2010),
24836:"There are no multiply-perfect Fibonacci numbers"
24415:Su, Francis E (2000), "Fibonacci GCD's, please",
24197:
23558:
23361: â Binary sequence from Fibonacci recurrence
22645:The Fibonacci number series is used for optional
22567:The keys in the left spine are Fibonacci numbers.
22256:without an odd number of consecutive integers is
21509:
21482:
18137:{\displaystyle {\bigl (}{\tfrac {p}{5}}{\bigr )}}
18094:The divisibility of Fibonacci numbers by a prime
16921:. Thus the Fibonacci sequence is an example of a
16458:
15806:
15747:
15731:
15603:
15353:
13973:
13960:
13818:
13805:
10924:. More precisely, this sequence corresponds to a
10515:{\displaystyle \sum _{i=1}^{n}F_{2i}=F_{2n+1}-1.}
10052:{\displaystyle \{(1,1,...,1,2)\},\{(1,1,...,1)\}}
9771:-th Fibonacci number minus 1. In symbols:
9108:{\displaystyle F_{4}=3=|\{(1,1,1),(1,2),(2,1)\}|}
7331:
7318:
7127:
7100:
7003:
6976:
6931:
6904:
6878:
6845:
6742:
6540:
6452:
6321:
6293:
6284:
6195:
6182:
6173:
5805:
5792:
5739:
5711:
5683:
5670:
5624:
5539:
5318:
5285:
5237:
5198:
3543:
2778:{\displaystyle {\frac {\varphi ^{n}}{\sqrt {5}}}}
1705:satisfy the Fibonacci recursion. In other words,
970:The Fibonacci sequence first appears in the book
28893:
25720:"The Fibonacci sequence as it appears in nature"
25612:Prusinkiewicz, Przemyslaw; Hanan, James (1989),
25468:Jones, Judy; Wilson, William (2006), "Science",
25403:
24863:
17990:is one greater or one less than a prime number.
17207:
17166:
17125:
16991:
16932:
11919:This can be used to prove Fibonacci identities.
11484:{\displaystyle \sum _{i=1}^{n+1}F_{i}=F_{n+3}-1}
10436:{\displaystyle \sum _{i=0}^{n-1}F_{2i+1}=F_{2n}}
7470:This property can be understood in terms of the
3771:
3585:
911:also expresses knowledge of the sequence in the
16:Numbers obtained by adding the two previous ones
26418:
25861:Journal of Optimization Theory and Applications
25191:, Cambridge Univ. Press, p. 121, Ex 1.35,
23777:"Fibonacci's Liber Abaci (Book of Calculation)"
23448:
23446:
23444:
22631:Fibonacci numbers arise in the analysis of the
22338:The Fibonacci numbers are also an example of a
19001:Such primes (if there are any) would be called
17542:These cases can be combined into a single, non-
11230:{\displaystyle \sum _{i=1}^{n}F_{i}=F_{n+2}-1.}
11089:{\displaystyle \sum _{i=0}^{\infty }F_{i}z^{i}}
3445:there are either 4 or 5 Fibonacci numbers with
1074:
26212:
26198:
26164:sequence A000045 (Fibonacci numbers)
25994:
25980:(3rd ed.), New Jersey: World Scientific,
25818:
25578:Formaliser le vivant - Lois, Théories, ModÚles
25293:, and Appendix B, The Ring Lemma, pp. 318â321.
24655:The Journal of the London Mathematical Society
24254:Glaister, P (1995), "Fibonacci power series",
24198:Flajolet, Philippe; Sedgewick, Robert (2009),
24173:
23013:, plus the number of male ancestors, which is
23001:, is the number of female ancestors, which is
20331:{\displaystyle F_{6}=8{\text{ and }}F_{7}=13.}
17931:2, 3, 5, 13, 89, 233, 1597, 28657, 514229, ...
17715:number – then we can calculate
10853:; from this the identity follows by comparing
10354:{\displaystyle \sum _{i=1}^{n}F_{i}=F_{n+2}-1}
9837:{\displaystyle \sum _{i=1}^{n}F_{i}=F_{n+2}-1}
3879:Since the golden ratio satisfies the equation
2523:{\displaystyle U_{n}=a\varphi ^{n}+b\psi ^{n}}
1910:{\displaystyle U_{n}=a\varphi ^{n}+b\psi ^{n}}
191:The Fibonacci spiral: an approximation of the
72:The Fibonacci numbers were first described in
28779:
28268:
28118:
26184:
25953:
25526:
24587:, Cambridge University Press, pp. 1â12,
24340:
23576:
22173:without consecutive integers, that is, those
21912:
21899:
21887:
21874:
21862:
21849:
20638:
20619:
20571:
20552:
20512:
20493:
20253:
20234:
20183:
20164:
20124:
20105:
19946:{\displaystyle F_{5}=5{\text{ and }}F_{6}=8.}
19868:
19849:
19801:
19782:
19742:
19723:
19558:{\displaystyle F_{3}=2{\text{ and }}F_{4}=3.}
19477:
19458:
19407:
19388:
19348:
19329:
19218:
19199:
19109:
19090:
18913:It is not known whether there exists a prime
18829:
18810:
18733:
18714:
18665:
18646:
18569:
18550:
18473:
18454:
18129:
18110:
5627:
5603:
5542:
5518:
4456:{\displaystyle \psi ^{n}=F_{n}\psi +F_{n-1}.}
3268:{\displaystyle \ln(\varphi )=0.481211\ldots }
370:is omitted, so that the sequence starts with
28715:Hypergeometric function of a matrix argument
28020:
26370:
25426:
24786:Rendiconti del Circolo Matematico di Palermo
24522:. Williams calls this property "well known".
24224:
24222:
24220:
23864:Faculty of Engineering and Physical Sciences
23655:(Hn.). Varanasi-I: TheChowkhamba Vidyabhawan
23610:, vol. 1, Addison Wesley, p. 100,
23441:
21124:are collected at the relevant repositories.
16914:-th number of the sequence is a multiple of
14376:{\displaystyle s(z)={\frac {z}{1-z-z^{2}}}.}
10255:... where the last two terms have the value
10234:
10198:
10182:
10140:
10046:
10010:
10004:
9962:
9541:
9481:
9465:
9405:
9267:
9147:
9097:
9037:
8987:
8957:
8907:
8895:
8845:
8836:
8786:
8783:
7618:are ratios of successive Fibonacci numbers:
1096:linear recurrence with constant coefficients
999:: how many pairs will there be in one year?
203:The Fibonacci numbers may be defined by the
28571:1 + 1/2 + 1/3 + ... (Riemann zeta function)
26021:
25467:
25358:
25333:Proceedings of the USSR Academy of Sciences
24994:
24982:
24970:
23452:
21407:The Fibonacci numbers occur as the sums of
11995:note that the left hand side multiplied by
11988:{\textstyle \sum _{i=1}^{n}F_{i}=F_{n+2}-1}
10913:{\displaystyle (F_{n})_{n\in \mathbb {N} }}
8543:
971:
953:
89:
28786:
28772:
28275:
28261:
28125:
28111:
26485:Possessing a specific set of other numbers
26308:
26191:
26177:
26024:Reciprocity Laws: From Euler to Eisenstein
25263:
25077:
24857:
24827:
24628:
24583:; Cooper, Joshua; Hurlbert, Glenn (eds.),
24368:
24204:, Cambridge University Press, p. 42,
22613:whose child trees (recursively) differ in
20787:
20786:
20780:
20779:
20405:
20404:
20398:
20397:
20017:
20016:
20010:
20009:
19632:
19631:
19625:
19624:
17687:is a prime, and if it fails to hold, then
17641:
17571:
16872:
16867:
15592:
10997:-th Fibonacci number equals the number of
10860:
7463:of â1, and thus it is a 2 Ă 2
3874:
2665:
1064:th month, the number of pairs is equal to
28627:1 + 1/2 + 1/3 + 1/4 + ⯠(harmonic series)
27948:
26895:
25880:
25755:
25491:
25289:; see especially Lemma 8.2 (Ring Lemma),
25210:
25123:On-Line Encyclopedia of Integer Sequences
24943:
24930:On-Line Encyclopedia of Integer Sequences
24719:
24555:On-Line Encyclopedia of Integer Sequences
24508:
24498:
24396:
24327:On-Line Encyclopedia of Integer Sequences
24217:
24029:On-Line Encyclopedia of Integer Sequences
24004:On-Line Encyclopedia of Integer Sequences
23691:
23495:
23459:, Indiana University Press, p. 126,
23434:On-Line Encyclopedia of Integer Sequences
22936:, the nearest neighbors of floret number
21929:, where we are choosing the positions of
21156:. The lengths of the periods for various
17731:efficiently using the matrix form. Thus
17645:
17575:
17023:
15283:{\textstyle z\mapsto -s\left(-1/z\right)}
10904:
8610:. This can be taken as the definition of
6461:
6205:
6201:
5186:that describes the Fibonacci sequence is
5157:
3424:
1380:
1275:
1085:
28282:
25954:Beck, Matthias; Geoghegan, Ross (2010),
25527:Scott, T.C.; Marketos, P. (March 2014),
24777:
24568:
24441:
24253:
24139:
23802:
23705:
23649:
23382:
23230:appears on all lines of the genealogy.)
23027:
22789:
22709:
22556:
22540:, used to prove connections between the
21946:
21394:
21232:using a modification of Binet's formula.
13735:These can be found experimentally using
12688:Catalan's identity is a generalization:
10928:. The specification of this sequence is
3914:{\displaystyle \varphi ^{2}=\varphi +1,}
3863:
3757:
3459:representation, the number of digits in
3374:{\displaystyle \varphi ^{n}/{\sqrt {5}}}
1051:
943:
793:
337:Under some older definitions, the value
186:
152:
59:
28841:Greedy algorithm for Egyptian fractions
25574:
25538:MacTutor History of Mathematics archive
25530:On the Origin of the Fibonacci Sequence
25186:
24047:, Oxford University Press, p. 92,
23874:
23594:
22572:The Fibonacci numbers are important in
22275:The number of binary strings of length
22210:The number of binary strings of length
22111:The number of binary strings of length
21963:as an ordered sum of 1s and 2s (called
21951:Use of the Fibonacci sequence to count
21127:
21106:All known factors of Fibonacci numbers
17315:is congruent to 2 or 3 modulo 5, then,
13107:. The last is an identity for doubling
12560:{\textstyle \varphi -\psi ={\sqrt {5}}}
11851:
11143:{\displaystyle {\frac {z}{1-z-z^{2}}}.}
10846:{\displaystyle F_{n},F_{n-1},...,F_{1}}
10615:A different trick may be used to prove
9288:In this manner the recurrence relation
8998:{\displaystyle F_{3}=2=|\{(1,1),(2)\}|}
8548:
4762:{\displaystyle {\sqrt {5}}\varphi ^{n}}
28894:
28056:
26079:
25751:
25749:
25136:
25044:Factors of Fibonacci and Lucas numbers
24080:
24042:
23807:, New York: Sterling, pp. 20â21,
23763:
23735:"A History of Piáč
gala's Combinatorics"
23708:'VáčttajÄtisamuccaya' of kavi Virahanka
23552:
23418:
23416:
22901:is the index number of the floret and
22310:without an even number of consecutive
22279:without an even number of consecutive
17295:is congruent to 1 or 4 modulo 5, then
16639:Moreover, this number has been proved
14315:and its sum has a simple closed form:
14074:
12808:
10782:and decompose it into squares of size
9948:{\displaystyle (2,...),(1,2,...),...,}
4544:{\displaystyle F_{n}=F_{n+2}-F_{n+1}.}
3974:yield Fibonacci numbers as the linear
3438:. As a consequence, for every integer
2789:, using the nearest integer function:
1690:{\displaystyle x^{n}=x^{n-1}+x^{n-2},}
1441:, this formula can also be written as
816:) end with a short syllable and five (
255:{\displaystyle F_{0}=0,\quad F_{1}=1,}
28767:
28256:
28106:
28055:
28019:
27983:
27947:
27907:
27532:
27421:
27147:
27062:
27017:
26894:
26584:
26531:
26483:
26417:
26369:
26307:
26211:
26172:
26067:
26040:
25914:
25902:
25790:
25665:
25639:
25562:
25550:
25302:
25237:
25189:Enumerative Combinatorics I (2nd ed.)
25174:
24686:
24631:AMS Dolciani Mathematical Expositions
24229:
24161:
23826:
23720:
23671:
23637:
23600:
23564:
23512:
23481:
23262:-th Fibonacci number. (However, when
22974:), according to the following rules:
22596:. A tape-drive implementation of the
22272:is to replace 1 with 0 and 2 with 11.
22240:without an odd number of consecutive
22214:without an odd number of consecutive
21070:
20988:
18035:With the exceptions of 1, 8 and 144 (
14385:This can be proved by multiplying by
9346:{\displaystyle F_{n}=F_{n-1}+F_{n-2}}
3749:{\displaystyle U_{1}=-U_{0}/\varphi }
477:{\displaystyle F_{n}=F_{n-1}+F_{n-2}}
320:{\displaystyle F_{n}=F_{n-1}+F_{n-2}}
28867:Generalizations of Fibonacci numbers
28862:List of things named after Fibonacci
28857:Fibonacci numbers in popular culture
26585:
26117:Periods of Fibonacci Sequences Mod m
25972:
25930:
24949:The New Book of Prime Number Records
24870:Annales Mathematicae at Informaticae
24783:
24652:
24476:{\displaystyle F_{p-\varepsilon }/p}
23980:
23968:
23732:
23588:
23354:Fibonacci numbers in popular culture
21198:Generalizations of Fibonacci numbers
17711:is large – say a 500-
17611:
13111:; other identities of this type are
9729:sequences, showing this is equal to
7474:representation for the golden ratio
4465:These expressions are also true for
1434:{\displaystyle \psi =-\varphi ^{-1}}
884:) is obtained by adding one to the
80:on enumerating possible patterns of
28592:1 â 1 + 1 â 1 + ⯠(Grandi's series)
27984:
25746:
24757:
23413:
22624:Fibonacci numbers are used by some
22552:
22009:rectangle). For example, there are
21692:
20830:
20771:
20448:
20389:
20060:
20001:
19678:
19616:
19274:
19267:
19242:
19235:
19165:
19158:
19133:
19126:
18971:
18418:
18358:
18278:
18271:
18233:
18226:
17908:
17870:
17616:The above formula can be used as a
17485:
17478:
17423:
17416:
15290:is the generating function for the
13739:, and are useful in setting up the
12570:
11152:
10775:{\displaystyle F_{n}\times F_{n+1}}
10612:-th Fibonacci number minus 1.
9611:and the cardinality of each set is
6825:The closed-form expression for the
1590:are both solutions of the equation
1106:, named after French mathematician
130:the arrangement of leaves on a stem
13:
28132:
27908:
25756:Hutchison, Luke (September 2004),
25581:(in French), Hermann, p. 28,
25009:Fibonacci and Lucas factorizations
24951:, New York: Springer, p. 64,
24414:
24045:A New Year Gift: On Hexagonal Snow
22794:Illustration of Vogel's model for
21903:
21878:
21853:
21707:
21486:
21191:
16676:
16593:
16543:
16502:
16280:
16142:
16001:
15901:
15687:
15546:
15510:
14854:
14749:
14699:
14655:
14598:
14548:
14504:
14308:{\displaystyle |z|<1/\varphi ,}
14245:This series is convergent for any
14126:
13964:
13809:
12579:Cassini's and Catalan's identities
11061:
10957:
10953:
10946:
10867:
9353:may be understood by dividing the
7322:
7104:
6980:
6951:
6908:
6849:
6695:
6645:
6601:
6297:
6186:
5796:
5715:
5674:
5289:
5202:
3781:
3595:
3006:
3003:
3000:
2997:
2994:
2991:
2988:
829:The Fibonacci sequence appears in
14:
28918:
28710:Generalized hypergeometric series
26098:
25516:, Computer Science For Fun: CS4FN
25510:"Marks for the da Vinci Code: Bâ"
25472:, Ballantine Books, p. 544,
25375:Amiga ROM Kernel Reference Manual
25213:"Review of Yuri V. Matiyasevich,
25080:The American Mathematical Monthly
24901:
24760:"On triangular Fibonacci numbers"
24574:"Probabilizing Fibonacci numbers"
23853:
18030:
17980:No Fibonacci number greater than
14083:of the Fibonacci sequence is the
8918:{\displaystyle F_{2}=1=|\{(1)\}|}
7842:arithmetic operations, using the
7831:, this matrix can be computed in
5182:A 2-dimensional system of linear
4553:
4391:{\displaystyle \psi ^{2}=\psi +1}
4352:{\displaystyle \psi =-1/\varphi }
2285:satisfy the system of equations:
1110:, though it was already known by
159:-th Fibonacci number in terms of
28748:
28747:
28720:Lauricella hypergeometric series
28438:
28241:
28086:
27694:Perfect digit-to-digit invariant
27063:
25692:The Algorithmic Beauty of Plants
25143:-step Fibonacci sequence modulo
23827:Knott, Ron (25 September 2016),
22913:, approximately 137.51°, is the
22536:Fibonacci numbers appear in the
21332:, the result is the sequence of
17699:and satisfies the formula, then
14068:
8856:{\displaystyle F_{1}=1=|\{()\}|}
5642:corresponding to the respective
5439:
5370:
2785:. Therefore, it can be found by
2448:producing the required formula.
28730:Riemann's differential equation
25812:
25796:
25737:
25712:
25671:
25633:
25605:
25568:
25520:
25502:
25485:
25461:
25420:
25404:Dean Leffingwell (2021-07-01),
25397:
25381:
25367:
25352:
25323:
25308:The Art of Computer Programming
25296:
25257:
25231:
25204:
25180:
25130:
25105:
25071:
25035:
25000:
24937:
24912:
24895:
24751:
24699:
24680:
24646:
24622:
24562:
24537:
24525:
24435:
24408:
24390:
24362:
24334:
24305:
24289:
24247:
24191:
24167:
24133:
24109:
24074:
24060:
24036:
24011:
23986:
23930:
23910:Discrete Mathematics with Ducks
23900:
23868:
23847:
23820:
23796:
23769:
23726:
23699:
23665:
23643:
23607:The Art of Computer Programming
23519:The Art of Computer Programming
23389:
22723:arrangement of leaves on a stem
22603:The Art of Computer Programming
22574:computational run-time analysis
22268:. A bijection with the sums to
21385:
21206:, and specifically by a linear
20823:
20764:
20596:
20529:
20441:
20382:
20211:
20141:
20053:
19994:
19826:
19759:
19671:
19609:
19435:
19365:
18964:
18411:
18372:
18366:
18351:
18144:which is evaluated as follows:
17863:
12589:Cassini's identity states that
11387:and so we have the formula for
10977:. Indeed, as stated above, the
10926:specifiable combinatorial class
9722:{\displaystyle F_{n-1}+F_{n-2}}
6754:
5692:
3571:{\displaystyle \varphi \colon }
2867:grows, being less than 0.1 for
2420:
1919:satisfies the same recurrence,
1848:It follows that for any values
1098:, the Fibonacci numbers have a
962:Biblioteca Nazionale di Firenze
494:The first 20 Fibonacci numbers
232:
25924:
25436:Journal of Theoretical Biology
25270:, Cambridge University Press,
25049:collects all known factors of
25014:collects all known factors of
24973:, pp. 73â74, ex. 2.25â28.
24486:Canadian Mathematical Bulletin
23907:Sarah-Marie Belcastro (2018).
23453:Goonatilake, Susantha (1998),
22626:pseudorandom number generators
22460:
22421:
22409:
22379:
21673:
21660:
21626:
21613:
21597:
21585:
21411:in the "shallow" diagonals of
21390:
21235:Starting with other integers.
21180:. However, for any particular
20834:
20824:
20775:
20765:
20452:
20442:
20393:
20383:
20064:
20054:
20005:
19995:
19682:
19672:
19620:
19610:
19278:
19268:
19246:
19236:
19169:
19159:
19137:
19127:
18982:
18965:
18422:
18412:
18362:
18352:
18282:
18272:
18237:
18227:
18004:1, 3, 21, and 55 are the only
17923:is a Fibonacci number that is
17874:
17864:
17691:is definitely not a prime. If
17495:
17489:
17479:
17433:
17427:
17417:
17377:
17248:
17210:
17201:
17169:
17160:
17128:
17018:
16994:
16980:
16935:
16258:
16252:
16220:
16207:
16198:
16185:
16019:
16009:
15457:
15451:
15428:
15422:
15344:
15338:
15310:
15304:
15247:
15031:
15025:
15013:partial fraction decomposition
14910:
14859:
14478:
14472:
14466:
14441:
14417:
14392:
14333:
14327:
14284:
14276:
14104:
14098:
13245:
13235:
12887:
12877:
12762:
12752:
12663:
12653:
12585:Cassini and Catalan identities
12482:
12457:
12437:
12425:
10964:
10941:
10893:
10879:
10238:
10231:
10201:
10194:
10186:
10179:
10143:
10136:
10043:
10013:
10001:
9965:
9927:
9903:
9897:
9879:
9545:
9526:
9508:
9502:
9484:
9477:
9469:
9450:
9432:
9426:
9408:
9401:
9271:
9264:
9252:
9246:
9228:
9222:
9204:
9198:
9180:
9174:
9150:
9143:
9101:
9094:
9082:
9076:
9064:
9058:
9040:
9033:
8991:
8984:
8978:
8972:
8960:
8953:
8911:
8904:
8898:
8891:
8849:
8842:
8839:
8832:
8797:{\displaystyle F_{0}=0=|\{\}|}
8790:
8779:
8727:
8711:
8485:
8450:
8427:
8392:
8369:
8331:
7870:
7860:
7614:of the continued fraction for
7407:
7397:
7234:
7224:
7064:
7054:
6079:
6061:
6051:
6030:
5986:
5949:
5902:
5859:
5832:
5774:
5699:
5658:
5456:
5422:
5381:
5348:
5177:
5148:
5112:
5100:
5090:
5031:
4976:
4889:
4879:
4710:
4688:
4678:
4662:
4255:
4223:
4195:
4183:
4119:
4084:
3778:
3592:
3544:Limit of consecutive quotients
3301:
3295:
3253:
3247:
3221:
3215:
3194:
3188:
3169:
3163:
3149:
3143:
3131:
3125:
3069:
3049:
3018:
3012:
2901:
2895:
2076:
2038:
2029:
1991:
1538:
1528:
1489:
1479:
412:{\displaystyle F_{1}=F_{2}=1,}
128:, such as branching in trees,
76:as early as 200 BC in work by
19:For the chamber ensemble, see
1:
28793:
28725:Modular hypergeometric series
28566:1/4 + 1/16 + 1/64 + 1/256 + âŻ
26533:Expressible via specific sums
25803:"Zeckendorf representation",
25427:Douady, S; Couder, Y (1996),
25211:Harizanov, Valentina (1995),
23377:
23279:for financial market trading.
22660:computers. The number series
22103:, or equivalently, among the
21742:and collecting like terms of
19015:is an odd prime number then:
18019:No Fibonacci number can be a
18008:Fibonacci numbers, which was
17683:, then this is evidence that
17331:. The remaining case is that
16910:) and, more generally, every
16477:reciprocal Fibonacci constant
12524:{\textstyle \varphi \psi =-1}
12499:as required, using the facts
11528:{\displaystyle {F_{n+1}}^{2}}
10920:is also considered using the
10288:. From this it follows that
4481:extended to negative integers
4398:and it is also the case that
3756:. This can be verified using
1056:Solution to Fibonacci rabbit
861:
182:
21:Fibonacci Sequence (ensemble)
25978:A Walk Through Combinatorics
25695:, Springer-Verlag, pp.
25654:10.1016/0025-5564(79)90080-4
25347:Soviet Mathematics - Doklady
25264:Stephenson, Kenneth (2005),
25137:LĂŒ, Kebo; Wang, Jun (2006),
24421:, et al, HMC, archived from
24299:
24178:, BirkhĂ€user, pp. 5â6,
24081:Gessel, Ira (October 1972),
24068:Strena seu de Nive Sexangula
23693:10.1016/0315-0860(85)90021-7
23497:10.1016/0315-0860(85)90021-7
23406:
22823:in 1979. This has the form
22753:form of some flowers. Field
22561:Fibonacci tree of height 6.
22372:, obtained from the formula
21145:, the resulting sequence is
20854:, all odd prime divisors of
11036:ordinary generating function
4793:{\displaystyle \varphi ^{n}}
3943:{\displaystyle \varphi ^{n}}
3326:
2266:then the resulting sequence
1108:Jacques Philippe Marie Binet
1075:Relation to the golden ratio
7:
28735:Theta hypergeometric series
27622:Multiplicative digital root
26149:Encyclopedia of Mathematics
26022:Lemmermeyer, Franz (2000),
24985:, pp. 73â74, ex. 2.28.
24738:10.4007/annals.2006.163.969
23947:10.1007/978-3-322-85165-9_6
23341:
22804:A model for the pattern of
18073:). As a result, 8 and 144 (
12567:to simplify the equations.
12012:{\displaystyle {\sqrt {5}}}
11922:For example, to prove that
11038:of the Fibonacci sequence,
4359:, it is also the case that
2451:Taking the starting values
825:) end with a long syllable.
790:Golden ratio § History
140:, and the arrangement of a
10:
28923:
28617:Infinite arithmetic series
28561:1/2 + 1/4 + 1/8 + 1/16 + âŻ
28556:1/2 â 1/4 + 1/8 â 1/16 + âŻ
27018:
26008:, Wiley, pp. 91â101,
25937:Princeton University Press
25540:, University of St Andrews
25113:Sloane, N. J. A.
24920:Sloane, N. J. A.
24545:Sloane, N. J. A.
24313:Sloane, N. J. A.
24019:Sloane, N. J. A.
23994:Sloane, N. J. A.
23424:Sloane, N. J. A.
23275:levels are widely used in
22783:, specifically as certain
22706:Golden ratio § Nature
22703:
22697:
22656:audio file format used on
22640:Fibonacci search technique
22617:by exactly 1. So it is an
22218:s is the Fibonacci number
22119:s is the Fibonacci number
21228:Generalizing the index to
21195:
21134:
17912:
15490:0.001001002003005008013021
14955:where all terms involving
13741:special number field sieve
12582:
10999:combinatorial compositions
7844:exponentiation by squaring
4472:if the Fibonacci sequence
3381:, the number of digits in
1856:, the sequence defined by
1078:
787:
778:
107:Fibonacci search technique
18:
28872:The Fibonacci Association
28849:
28828:
28801:
28743:
28700:
28644:
28579:
28548:
28541:
28511:
28480:
28473:
28447:
28436:
28359:
28303:
28294:
28239:
28140:
28082:
28065:
28051:
28029:
28015:
27993:
27979:
27957:
27943:
27916:
27903:
27879:
27833:
27793:
27744:
27718:
27699:Perfect digital invariant
27651:
27635:
27614:
27581:
27546:
27542:
27528:
27436:
27417:
27386:
27353:
27310:
27287:
27274:Superior highly composite
27164:
27160:
27143:
27071:
27058:
27026:
27013:
26901:
26890:
26852:
26843:
26821:
26778:
26740:
26731:
26664:
26606:
26597:
26593:
26580:
26538:
26527:
26490:
26479:
26427:
26413:
26376:
26365:
26318:
26303:
26221:
26207:
25873:10.1007/s10957-012-0061-2
25679:Prusinkiewicz, Przemyslaw
25187:Stanley, Richard (2011),
24369:Honsberger, Ross (1985),
24043:Kepler, Johannes (1966),
23653:PÄáčinikÄlÄ«na BhÄratavaráčŁa
23577:Beck & Geoghegan 2010
23365:Random Fibonacci sequence
23348:The Fibonacci Association
23335:Golden ratio § Music
22990:ancestors at each level,
22737:, and the family tree of
22693:
22642:, uses Fibonacci numbers.
22161:is the number of subsets
22002:{\displaystyle 2\times n}
21149:with period at most
10059:each with cardinality 1.
4483:using the Fibonacci rule
2874:, and less than 0.01 for
1102:. It has become known as
939:
132:, the fruit sprouts of a
27312:Euler's totient function
27096:EulerâJacobi pseudoprime
26371:Other polynomial numbers
25839:10.1109/tim.1985.4315428
25642:Mathematical Biosciences
25575:Varenne, Franck (2010),
25410:, Scaled Agile Framework
24667:10.1112/jlms/s1-39.1.537
24298:(1899) quoted according
24256:The Mathematical Gazette
23803:Hemenway, Priya (2005),
23233:
22777:PrzemysĆaw Prusinkiewicz
21379:n-Step Fibonacci numbers
18016:and proved by Luo Ming.
15172:is the golden ratio and
14423:{\textstyle (1-z-z^{2})}
11164:For example, reconsider
10555:{\displaystyle F_{2n-1}}
9955:until the last two sets
8544:Combinatorial identities
7652:-th convergent, and the
5760:As the initial value is
4631:{\displaystyle 5x^{2}-4}
4595:{\displaystyle 5x^{2}+4}
3972:recurrence relationships
3963:{\displaystyle \varphi }
3852:{\displaystyle \varphi }
1094:defined by a homogenous
783:
28448:Properties of sequences
27126:SomerâLucas pseudoprime
27116:LucasâCarmichael number
26951:Lazy caterer's sequence
26068:Lucas, Ădouard (1891),
25727:The Fibonacci Quarterly
25470:An Incomplete Education
24997:, p. 73, ex. 2.27.
24302:, Page 95, Exercise 3b.
24091:The Fibonacci Quarterly
24083:"Fibonacci is a Square"
23743:Northeastern University
23456:Toward a Global Science
23331:a system of composition
23217:{\displaystyle F_{5}=5}
23180:{\displaystyle F_{4}=3}
23143:{\displaystyle F_{3}=2}
23106:{\displaystyle F_{2}=1}
23071:{\displaystyle F_{1}=1}
22759:Karl Friedrich Schimper
22733:, the arrangement of a
22582:greatest common divisor
22333:Hilbert's tenth problem
21953:{1, 2}-restricted
18312:is a prime number then
17108:{\displaystyle F_{2}=1}
17075:{\displaystyle F_{1}=1}
17040:greatest common divisor
16903:{\displaystyle F_{3}=2}
16873:Divisibility properties
16868:Primes and divisibility
15001:{\displaystyle k\geq 2}
13274:by Cassini's identity.
11263:{\displaystyle F_{n+1}}
10281:{\displaystyle F_{1}=1}
9670:{\displaystyle F_{n-2}}
9637:{\displaystyle F_{n-1}}
8734:{\displaystyle |{...}|}
8696:{\displaystyle F_{1}=1}
8663:{\displaystyle F_{0}=0}
8555:combinatorial arguments
4735:, can be multiplied by
3875:Decomposition of powers
3452:More generally, in the
2666:Computation by rounding
1086:Closed-form expression
980:The Book of Calculation
363:{\displaystyle F_{0}=0}
28311:Arithmetic progression
27001:WedderburnâEtherington
26401:Lucky numbers of Euler
26080:Sigler, L. E. (2002),
25958:, New York: Springer,
25825:, IM-34 (4): 650â653,
25448:10.1006/jtbi.1996.0026
25377:, AddisonâWesley, 1991
25215:Hibert's Tenth Problem
24500:10.4153/CMB-1982-053-0
24477:
24402:My Numbers, My Friends
24321:-th Fibonacci number)"
24201:Analytic Combinatorics
24146:In honour of Fibonacci
23782:The University of Utah
23329:(1895â1943) developed
23218:
23181:
23144:
23107:
23072:
23034:
22889:
22801:
22718:
22677:Scaled Agile Framework
22609:A Fibonacci tree is a
22568:
22542:circle packing theorem
22507:
22003:
21956:
21923:
21763:
21736:
21711:
21522:
21478:
21404:
21176:or of an element in a
21098:
20842:
20717:
20664:
20460:
20332:
20279:
20072:
19947:
19894:
19690:
19559:
19506:
19296:
18993:
18905:
18433:
18300:
18138:
17899:modular exponentiation
17891:Here the matrix power
17885:
17673:
17603:
17534:
17261:
17109:
17076:
17028:
16904:
16859:
16774:
16737:
16680:
16631:
16597:
16547:
16506:
16467:
16325:
16284:
16232:
16146:
16111:
16056:
16005:
15962:
15905:
15870:
15691:
15656:
15550:
15502:
15435:
15406:
15384:
15317:
15284:
15224:
15166:
15108:
15002:
14976:
14947:
14858:
14753:
14703:
14659:
14602:
14552:
14508:
14424:
14377:
14309:
14262:
14237:
14130:
14052:
13956:
13897:
13801:
13729:
13523:
13397:
13268:
13082:
12916:
12800:
12682:
12561:
12525:
12493:
12013:
11989:
11949:
11913:
11843:
11788:
11750:
11644:
11611:
11562:
11529:
11485:
11445:
11407:
11378:
11300:
11264:
11231:
11191:
11159:mathematical induction
11144:
11090:
11065:
11025:
10991:
10971:
10914:
10847:
10776:
10724:is the product of the
10718:
10691:
10642:
10598:
10597:{\displaystyle F_{2n}}
10556:
10516:
10470:
10437:
10397:
10355:
10315:
10282:
10246:
10053:
9949:
9866:
9838:
9798:
9750:
9723:
9671:
9638:
9605:
9579:
9553:
9374:
9347:
9279:
9109:
8999:
8919:
8857:
8798:
8735:
8697:
8664:
8631:
8604:
8578:
8509:
8191:
7943:
7819:
7602:
7445:
7348:
6817:
6678:
6553:
6352:closed-form expression
6338:
5872:
5752:
5636:
5551:
5472:
5400:
5334:alternatively denoted
5328:
5165:
5046:
4913:
4794:
4763:
4729:
4632:
4596:
4562:is a Fibonacci number
4545:
4457:
4392:
4353:
4316:
4037:
3964:
3944:
3915:
3869:
3853:
3833:
3750:
3699:
3672:
3643:
3572:
3535:
3432:
3375:
3317:
3269:
3228:
3099:
2959:
2855:
2779:
2721:
2657:
2524:
2440:
2361:
2230:
1911:
1840:
1691:
1623:
1622:{\textstyle x^{2}=x+1}
1574:
1435:
1394:
1286:
1224:
1100:closed-form expression
1071:
1042:-th Fibonacci number.
1034:). The number in the
972:
967:
954:
931:
895:cases and one to the
826:
478:
413:
364:
321:
256:
200:
171:, which obey the same
136:, the flowering of an
126:in biological settings
90:
69:
28702:Hypergeometric series
28316:Geometric progression
27289:Prime omega functions
27106:Frobenius pseudoprime
26896:Combinatorial numbers
26765:Centered dodecahedral
26558:Primary pseudoperfect
25345:by Myron J. Ricci in
25240:Mathematics in School
24905:The Fibonacci numbers
24478:
24375:Mathematical Gems III
23733:Shah, Jayant (1991),
23706:Velankar, HD (1962),
23650:Agrawala, VS (1969),
23383:Explanatory footnotes
23273:Fibonacci retracement
23219:
23182:
23145:
23108:
23073:
23031:
22890:
22793:
22725:, the fruitlets of a
22713:
22698:Further information:
22560:
22508:
22004:
21950:
21924:
21764:
21762:{\displaystyle x^{n}}
21737:
21691:
21523:
21435:
21409:binomial coefficients
21398:
21334:Fibonacci polynomials
21099:
20843:
20718:
20665:
20461:
20333:
20280:
20073:
19948:
19895:
19691:
19560:
19507:
19297:
18994:
18906:
18434:
18301:
18139:
17927:. The first few are:
17886:
17705:Fibonacci pseudoprime
17674:
17620:in the sense that if
17604:
17535:
17262:
17110:
17077:
17029:
16923:divisibility sequence
16905:
16860:
16754:
16738:
16660:
16645:Richard André-Jeannin
16632:
16577:
16527:
16486:
16468:
16326:
16264:
16233:
16126:
16112:
16036:
15985:
15963:
15885:
15871:
15671:
15657:
15530:
15503:
15436:
15407:
15385:
15318:
15285:
15237:The related function
15225:
15167:
15109:
15003:
14977:
14975:{\displaystyle z^{k}}
14948:
14838:
14733:
14683:
14639:
14582:
14532:
14488:
14425:
14378:
14310:
14263:
14238:
14110:
14053:
13936:
13898:
13781:
13730:
13524:
13398:
13269:
13083:
12917:
12801:
12683:
12562:
12526:
12494:
12014:
11990:
11929:
11914:
11856:The Binet formula is
11844:
11762:
11751:
11624:
11612:
11542:
11530:
11486:
11419:
11408:
11379:
11280:
11265:
11232:
11171:
11145:
11091:
11045:
11031:using terms 1 and 2.
11026:
10992:
10972:
10915:
10848:
10777:
10719:
10717:{\displaystyle F_{n}}
10692:
10622:
10599:
10557:
10517:
10450:
10438:
10371:
10356:
10295:
10283:
10247:
10054:
9950:
9867:
9839:
9778:
9751:
9749:{\displaystyle F_{n}}
9724:
9672:
9639:
9606:
9580:
9554:
9375:
9373:{\displaystyle F_{n}}
9348:
9280:
9110:
9000:
8920:
8858:
8799:
8736:
8698:
8665:
8637:with the conventions
8632:
8630:{\displaystyle F_{n}}
8605:
8579:
8577:{\displaystyle F_{n}}
8510:
8192:
7944:
7820:
7603:
7446:
7349:
6818:
6679:
6554:
6339:
5873:
5753:
5637:
5552:
5473:
5401:
5329:
5166:
5047:
4914:
4795:
4764:
4730:
4633:
4597:
4546:
4458:
4393:
4354:
4317:
4043:This equation can be
4038:
3970:and 1. The resulting
3965:
3945:
3916:
3867:
3854:
3834:
3751:
3700:
3698:{\displaystyle U_{1}}
3673:
3671:{\displaystyle U_{0}}
3644:
3573:
3536:
3433:
3376:
3318:
3270:
3229:
3100:
2960:
2856:
2780:
2722:
2658:
2525:
2441:
2362:
2231:
1912:
1841:
1692:
1624:
1575:
1436:
1395:
1287:
1225:
1055:
947:
923:
833:, in connection with
797:
479:
414:
365:
322:
257:
190:
63:
28682:Trigonometric series
28474:Properties of series
28321:Harmonic progression
27748:-composition related
27548:Arithmetic functions
27150:Arithmetic functions
27086:Elliptic pseudoprime
26770:Centered icosahedral
26750:Centered tetrahedral
25996:Borwein, Jonathan M.
25806:Encyclopedia of Math
25683:Lindenmayer, Aristid
25349:, 3:1259â1263, 1962.
25151:Utilitas Mathematica
24446:
23862:University of Surrey
23834:University of Surrey
23675:Historia Mathematica
23484:Historia Mathematica
23195:
23158:
23121:
23084:
23049:
22829:
22785:Lindenmayer grammars
22765:discovered that the
22688:Negafibonacci coding
22598:polyphase merge sort
22376:
22351:Zeckendorf's theorem
22326:Diophantine equation
22141:without consecutive
22115:without consecutive
21987:
21842:
21746:
21532:
21419:
21170:multiplicative order
20883:
20726:
20673:
20488:
20341:
20288:
20100:
19956:
19903:
19718:
19568:
19515:
19324:
19019:
18923:
18445:
18316:
18148:
18105:
18071:Carmichael's theorem
17993:The only nontrivial
17897:is calculated using
17737:
17624:
17554:
17355:
17122:
17086:
17053:
16929:
16881:
16751:
16657:
16483:
16335:
16245:
16123:
15982:
15882:
15668:
15527:
15445:
15434:{\displaystyle s(z)}
15416:
15396:
15332:
15316:{\displaystyle s(z)}
15298:
15241:
15176:
15118:
15019:
14986:
14959:
14434:
14389:
14321:
14272:
14252:
14092:
13911:
13756:
13747:a Fibonacci number.
13532:
13406:
13280:
13115:
12925:
12816:
12692:
12593:
12535:
12503:
12023:
11999:
11926:
11860:
11852:Binet formula proofs
11759:
11621:
11539:
11497:
11416:
11391:
11277:
11270:to both sides gives
11241:
11168:
11103:
11042:
11034:It follows that the
11009:
10981:
10932:
10876:
10786:
10740:
10701:
10619:
10578:
10530:
10447:
10368:
10292:
10259:
10069:
9959:
9876:
9850:
9775:
9763:-th is equal to the
9733:
9681:
9648:
9615:
9589:
9563:
9384:
9357:
9292:
9120:
9010:
8930:
8868:
8809:
8756:
8707:
8674:
8641:
8614:
8588:
8561:
8557:using the fact that
8549:Combinatorial proofs
8319:
8213:
8199:In particular, with
7996:
7857:
7687:
7484:
7360:
6835:
6692:
6581:
6358:
5888:
5880:it follows that the
5764:
5649:
5561:
5495:
5412:
5338:
5192:
5184:difference equations
5059:
4925:
4810:
4777:
4739:
4646:
4606:
4570:
4487:
4402:
4363:
4326:
4062:
3982:
3954:
3927:
3883:
3843:
3767:
3709:
3682:
3655:
3581:
3559:
3474:
3396:
3346:
3279:
3238:
3109:
2979:
2889:
2793:
2753:
2674:
2538:
2475:
2375:
2291:
1925:
1862:
1711:
1633:
1594:
1447:
1406:
1312:
1238:
1124:
423:
374:
341:
266:
210:
26:In mathematics, the
28878:Fibonacci Quarterly
28818:The Book of Squares
28662:Formal power series
27674:Kaprekar's constant
27194:Colossally abundant
27081:Catalan pseudoprime
26981:SchröderâHipparchus
26760:Centered octahedral
26636:Centered heptagonal
26626:Centered pentagonal
26616:Centered triangular
26216:and related numbers
26144:"Fibonacci numbers"
26071:Théorie des nombres
25831:1985ITIM...34..650P
25495:Fibonacci Quarterly
25361:Fibonacci Quarterly
25343:English translation
24730:2004math......3046B
24418:Mudd Math Fun Facts
24141:Dijkstra, Edsger W.
23983:, pp. 155â156.
23939:Der Goldene Schnitt
23926:Extract of page 260
23880:Mathematical Circus
23857:Fibonacci's Rabbits
23766:, pp. 404â405.
22959:, which depends on
22729:, the flowering of
22565:green; heights red.
21290:Primefree sequences
21208:difference equation
21204:recurrence relation
21160:form the so-called
21128:Periodicity modulo
19003:WallâSunâSun primes
17903:adapted to matrices
17546:formula, using the
17338:, and in this case
16747:tends to infinity:
16653:gives the identity
15515:Infinite sums over
15325:functional equation
14081:generating function
14075:Generating function
12809:d'Ocagne's identity
11803:
11659:
11577:
11406:{\displaystyle n+1}
11024:{\displaystyle n-1}
10657:
9865:{\displaystyle n+1}
9604:{\displaystyle n-3}
9578:{\displaystyle n-2}
8603:{\displaystyle n-1}
8310:
7568:
7556:
7542:
7530:
7516:
7504:
7425:
7381:
7356:which again yields
6571:through use of its
6524:
6502:
6479:
6467:
6436:
6414:
6391:
6379:
6268:
6246:
6223:
6211:
6157:
6135:
6112:
6100:
2369:which has solution
2246:are chosen so that
419:and the recurrence
205:recurrence relation
195:created by drawing
173:recurrence relation
102:Fibonacci Quarterly
38:, commonly denoted
28836:Fibonacci sequence
28460:Monotonic function
28379:Fibonacci sequence
28165:Fibonacci sequence
28092:Mathematics portal
28034:Aronson's sequence
27780:SmarandacheâWellin
27537:-dependent numbers
27244:Primitive abundant
27131:Strong pseudoprime
27121:Perrin pseudoprime
27101:Fermat pseudoprime
27041:Wolstenholme prime
26865:Squared triangular
26651:Centered decagonal
26646:Centered nonagonal
26641:Centered octagonal
26631:Centered hexagonal
26128:Fibonacci Sequence
25565:, pp. 112â13.
24798:10.1007/BF02904236
24758:Luo, Ming (1989),
24510:10338.dmlcz/137492
24473:
24235:"Fibonacci Number"
24231:Weisstein, Eric W.
23785:, 13 December 2009
23723:, p. 197â198.
23555:, pp. 404â05.
23327:Joseph Schillinger
23277:technical analysis
23245:refractive indexes
23228:population founder
23214:
23177:
23140:
23103:
23068:
23035:
22885:
22802:
22719:
22700:Patterns in nature
22578:Euclid's algorithm
22569:
22531:parallel computing
22503:
22370:Pythagorean triple
21999:
21957:
21919:
21918:
21759:
21732:
21518:
21405:
21094:
21074:
20992:
20838:
20713:
20660:
20634:
20607:
20567:
20540:
20508:
20456:
20328:
20275:
20249:
20222:
20179:
20152:
20120:
20068:
19943:
19890:
19864:
19837:
19797:
19770:
19738:
19686:
19555:
19502:
19473:
19446:
19403:
19376:
19344:
19292:
19287:
19214:
19187:
19105:
19078:
18989:
18901:
18899:
18825:
18729:
18661:
18565:
18469:
18429:
18296:
18291:
18134:
18125:
18098:is related to the
17881:
17850:
17810:
17669:
17599:
17530:
17525:
17257:
17105:
17072:
17024:
16900:
16855:
16733:
16627:
16463:
16462:
16321:
16320:
16228:
16107:
15958:
15866:
15652:
15498:
15431:
15402:
15380:
15313:
15280:
15220:
15195:
15162:
15137:
15104:
14998:
14972:
14943:
14941:
14420:
14373:
14305:
14258:
14233:
14048:
13893:
13725:
13519:
13393:
13264:
13078:
12912:
12796:
12678:
12557:
12521:
12489:
12487:
12009:
11985:
11909:
11839:
11789:
11746:
11645:
11607:
11563:
11525:
11481:
11403:
11374:
11260:
11227:
11140:
11086:
11021:
10987:
10967:
10910:
10843:
10772:
10714:
10687:
10643:
10594:
10552:
10512:
10433:
10351:
10278:
10242:
10049:
9945:
9862:
9834:
9746:
9719:
9677:giving a total of
9667:
9634:
9601:
9575:
9549:
9370:
9343:
9275:
9105:
8995:
8915:
8853:
8794:
8731:
8693:
8660:
8627:
8600:
8574:
8505:
8503:
8187:
8185:
7939:
7851:Cassini's identity
7815:
7806:
7721:
7598:
7591:
7586:
7581:
7563:
7537:
7511:
7472:continued fraction
7441:
7434:
7420:
7344:
7342:
7309:
7248:
7186:
7078:
6813:
6804:
6745:
6674:
6672:
6573:eigendecomposition
6549:
6531:
6519:
6488:
6474:
6443:
6431:
6400:
6386:
6334:
6332:
6275:
6263:
6232:
6218:
6164:
6152:
6121:
6107:
5868:
5748:
5632:
5599:
5547:
5514:
5468:
5396:
5324:
5276:
5161:
5052:, it follows that
5042:
4921:Comparing this to
4909:
4790:
4771:quadratic equation
4759:
4725:
4628:
4592:
4541:
4453:
4388:
4349:
4312:
4033:
3960:
3940:
3911:
3870:
3849:
3829:
3785:
3746:
3695:
3668:
3639:
3599:
3568:
3531:
3428:
3371:
3313:
3265:
3224:
3095:
2967:Instead using the
2955:
2851:
2775:
2717:
2653:
2651:
2520:
2436:
2357:
2352:
2226:
2224:
1907:
1836:
1834:
1687:
1619:
1570:
1431:
1390:
1282:
1220:
1072:
1016:At the end of the
968:
831:Indian mathematics
827:
474:
409:
360:
317:
252:
201:
74:Indian mathematics
70:
28:Fibonacci sequence
28907:Integer sequences
28902:Fibonacci numbers
28889:
28888:
28761:
28760:
28692:Generating series
28640:
28639:
28612:1 â 2 + 4 â 8 + âŻ
28607:1 + 2 + 4 + 8 + âŻ
28602:1 â 2 + 3 â 4 + âŻ
28597:1 + 2 + 3 + 4 + âŻ
28587:1 + 1 + 1 + 1 + âŻ
28537:
28536:
28465:Periodic sequence
28434:
28433:
28419:Triangular number
28409:Pentagonal number
28389:Heptagonal number
28374:Complete sequence
28296:Integer sequences
28250:
28249:
28219:Supersilver ratio
28200:Supergolden ratio
28100:
28099:
28078:
28077:
28047:
28046:
28011:
28010:
27975:
27974:
27939:
27938:
27899:
27898:
27895:
27894:
27714:
27713:
27524:
27523:
27413:
27412:
27409:
27408:
27355:Aliquot sequences
27166:Divisor functions
27139:
27138:
27111:Lucas pseudoprime
27091:Euler pseudoprime
27076:Carmichael number
27054:
27053:
27009:
27008:
26886:
26885:
26882:
26881:
26878:
26877:
26839:
26838:
26727:
26726:
26684:Square triangular
26576:
26575:
26523:
26522:
26475:
26474:
26409:
26408:
26361:
26360:
26299:
26298:
26091:978-0-387-95419-6
26033:978-3-540-66957-9
26015:978-0-471-31515-5
26000:Borwein, Peter B.
25987:978-981-4335-23-2
25965:978-1-4419-7022-0
25946:978-0-691-11321-0
25935:, Princeton, NJ:
25793:, pp. 98â99.
25706:978-0-387-97297-8
25627:978-0-387-97092-9
25479:978-0-7394-7582-9
25317:978-0-201-89683-1
25277:978-0-521-82356-2
25198:978-1-107-60262-5
25126:, OEIS Foundation
24958:978-0-387-94457-9
24933:, OEIS Foundation
24714:(163): 969â1018,
24640:978-0-88385-318-4
24594:978-1-107-15398-1
24558:, OEIS Foundation
24404:, Springer-Verlag
24371:"Millin's series"
24330:, OEIS Foundation
24185:978-3-7643-6135-8
24176:Fibonacci Numbers
24054:978-0-19-858120-8
24032:, OEIS Foundation
24007:, OEIS Foundation
23956:978-3-8154-2511-4
23920:978-1-351-68369-2
23889:978-0-88385-506-5
23617:978-81-7758-754-8
23529:978-0-321-33570-8
23466:978-0-253-33388-9
23437:, OEIS Foundation
23297:golden ratio base
23251:reflections, for
22909:. The divergence
22883:
22868:
22858:
22808:in the head of a
22647:lossy compression
22600:was described in
22580:to determine the
22340:complete sequence
22322:Yuri Matiyasevich
22200:with the sums to
22149:. Equivalently,
22147:Fibbinary numbers
22084:
22083:
21910:
21885:
21860:
21834:
21833:
21564:
21507:
21471:
21413:Pascal's triangle
21401:Pascal's triangle
21055:
21033:
21011:
20973:
20951:
20929:
20907:
20784:
20695:
20633:
20606:
20566:
20539:
20507:
20402:
20310:
20248:
20221:
20178:
20151:
20119:
20014:
19925:
19863:
19836:
19796:
19769:
19737:
19629:
19537:
19472:
19445:
19402:
19375:
19343:
19255:
19213:
19186:
19146:
19104:
19077:
19047:
18949:
18824:
18728:
18660:
18564:
18468:
18396:
18370:
18344:
18256:
18211:
18188:
18163:
18124:
17975:composite numbers
17959:, so, apart from
17658:
17612:Primality testing
17588:
16850:
16797:
16728:
16722:
16703:
16616:
16572:
16522:
16451:
16410:
16367:
16357:
16315:
16175:
16165:
16102:
16090:
16077:
15953:
15949:
15938:
15839:
15833:
15780:
15774:
15729:
15716:
15636:
15630:
15590:
15586:
15575:
15484:
15471:
15405:{\displaystyle z}
15370:
15213:
15194:
15155:
15136:
15097:
15073:
15047:
15046:
14368:
14261:{\displaystyle z}
13971:
13905:or alternatively
13816:
13737:lattice reduction
12555:
12455:
12375:
12301:
12262:
12216:
12174:
12007:
11868:
11535:to both sides of
11135:
11098:rational function
10990:{\displaystyle n}
7658: + 1)
7593:
7588:
7583:
7567:
7555:
7541:
7529:
7515:
7503:
7465:unimodular matrix
7436:
7432:
7424:
7380:
7329:
7264:
7263:
7125:
7001:
6929:
6876:
6533:
6523:
6516:
6501:
6490:
6486:
6478:
6466:
6445:
6435:
6428:
6413:
6402:
6398:
6390:
6378:
6319:
6277:
6267:
6260:
6245:
6234:
6230:
6222:
6210:
6193:
6166:
6156:
6149:
6134:
6123:
6119:
6111:
6099:
6082:
6049:
6048:
6033:
6012:
6011:
5989:
5968:
5967:
5952:
5931:
5930:
5905:
5862:
5851:
5850:
5835:
5824:
5823:
5803:
5777:
5737:
5702:
5681:
5661:
5623:
5619:
5598:
5538:
5534:
5513:
5459:
5425:
5384:
5351:
5316:
5235:
4994:
4904:
4898:
4844:
4802:quadratic formula
4747:
4723:
3814:
3770:
3628:
3584:
3526:
3470:is asymptotic to
3392:is asymptotic to
3369:
3082:
3047:
2945:
2930:
2841:
2830:
2829:
2773:
2772:
2715:
2698:
2697:
2644:
2643:
2590:
2589:
2415:
2414:
2400:
1697:so the powers of
1565:
1507:
1506:
1369:
1341:
1335:
1267:
1261:
1215:
1214:
1178:
1112:Abraham de Moivre
1038:-th month is the
774:
773:
36:Fibonacci numbers
28914:
28788:
28781:
28774:
28765:
28764:
28751:
28750:
28677:Dirichlet series
28546:
28545:
28478:
28477:
28442:
28414:Polygonal number
28394:Hexagonal number
28367:
28301:
28300:
28277:
28270:
28263:
28254:
28253:
28245:
28127:
28120:
28113:
28104:
28103:
28090:
28053:
28052:
28022:Natural language
28017:
28016:
27981:
27980:
27949:Generated via a
27945:
27944:
27905:
27904:
27810:Digit-reassembly
27775:Self-descriptive
27579:
27578:
27544:
27543:
27530:
27529:
27481:LucasâCarmichael
27471:Harmonic divisor
27419:
27418:
27345:Sparsely totient
27320:Highly cototient
27229:Multiply perfect
27219:Highly composite
27162:
27161:
27145:
27144:
27060:
27059:
27015:
27014:
26996:Telephone number
26892:
26891:
26850:
26849:
26831:Square pyramidal
26813:Stella octangula
26738:
26737:
26604:
26603:
26595:
26594:
26587:Figurate numbers
26582:
26581:
26529:
26528:
26481:
26480:
26415:
26414:
26367:
26366:
26305:
26304:
26209:
26208:
26193:
26186:
26179:
26170:
26169:
26162:
26157:
26107:
26094:
26075:
26064:
26036:
26018:
25990:
25968:
25949:
25918:
25912:
25906:
25900:
25894:
25893:
25884:
25856:
25850:
25849:
25816:
25810:
25809:
25800:
25794:
25788:
25782:
25781:
25780:
25779:
25773:
25767:, archived from
25762:
25753:
25744:
25741:
25735:
25734:
25733:(1): 53â56, 1963
25724:
25716:
25710:
25709:
25675:
25669:
25663:
25657:
25656:
25637:
25631:
25630:
25609:
25603:
25602:
25596:
25595:
25572:
25566:
25560:
25554:
25548:
25542:
25541:
25535:
25524:
25518:
25517:
25506:
25500:
25499:
25489:
25483:
25482:
25465:
25459:
25458:
25456:
25450:, archived from
25433:
25424:
25418:
25417:
25416:
25415:
25401:
25395:
25394:
25385:
25379:
25378:
25371:
25365:
25364:
25356:
25350:
25341:
25327:
25321:
25320:
25300:
25294:
25288:
25261:
25255:
25254:
25235:
25229:
25228:
25208:
25202:
25201:
25184:
25178:
25172:
25166:
25165:
25146:
25142:
25134:
25128:
25127:
25109:
25103:
25102:
25075:
25069:
25067:
25059:
25048:
25039:
25033:
25031:
25024:
25013:
25004:
24998:
24995:Lemmermeyer 2000
24992:
24986:
24983:Lemmermeyer 2000
24980:
24974:
24971:Lemmermeyer 2000
24968:
24962:
24961:
24945:Ribenboim, Paulo
24941:
24935:
24934:
24916:
24910:
24909:
24899:
24893:
24892:
24861:
24855:
24854:
24831:
24825:
24824:
24781:
24775:
24774:
24767:Fibonacci Quart.
24764:
24755:
24749:
24748:
24723:
24703:
24697:
24696:
24684:
24678:
24677:
24650:
24644:
24643:
24626:
24620:
24619:
24618:
24617:
24611:
24605:, archived from
24578:
24566:
24560:
24559:
24541:
24535:
24529:
24523:
24521:
24512:
24502:
24482:
24480:
24479:
24474:
24469:
24464:
24463:
24439:
24433:
24432:
24431:
24430:
24412:
24406:
24405:
24398:Ribenboim, Paulo
24394:
24388:
24387:
24366:
24360:
24359:
24338:
24332:
24331:
24320:
24309:
24303:
24293:
24287:
24286:
24251:
24245:
24244:
24243:
24226:
24215:
24214:
24195:
24189:
24188:
24171:
24165:
24159:
24153:
24152:
24151:
24137:
24131:
24130:
24128:
24127:
24113:
24107:
24106:
24105:
24103:
24087:
24078:
24072:
24071:
24064:
24058:
24057:
24040:
24034:
24033:
24015:
24009:
24008:
23990:
23984:
23978:
23972:
23966:
23960:
23959:
23934:
23928:
23924:
23904:
23898:
23897:
23872:
23866:
23865:
23851:
23845:
23844:
23843:
23841:
23824:
23818:
23817:
23800:
23794:
23793:
23792:
23790:
23773:
23767:
23761:
23755:
23754:
23753:
23751:
23739:
23730:
23724:
23718:
23712:
23711:
23703:
23697:
23696:
23695:
23669:
23663:
23662:
23647:
23641:
23635:
23626:
23625:
23598:
23592:
23586:
23580:
23574:
23568:
23562:
23556:
23550:
23544:
23543:
23540:
23510:
23501:
23500:
23499:
23479:
23470:
23469:
23450:
23439:
23438:
23420:
23400:
23393:
23301:
23268:
23261:
23257:
23250:
23225:
23223:
23221:
23220:
23215:
23207:
23206:
23188:
23186:
23184:
23183:
23178:
23170:
23169:
23151:
23149:
23147:
23146:
23141:
23133:
23132:
23114:
23112:
23110:
23109:
23104:
23096:
23095:
23077:
23075:
23074:
23069:
23061:
23060:
23033:Relationships".)
23024:
23012:
23000:
22962:
22958:
22954:
22939:
22935:
22904:
22900:
22894:
22892:
22891:
22886:
22884:
22879:
22866:
22859:
22857:
22856:
22847:
22839:
22822:
22812:was proposed by
22800:
22715:Yellow chamomile
22683:Fibonacci coding
22595:
22553:Computer science
22527:network topology
22523:undirected graph
22512:
22510:
22509:
22504:
22499:
22498:
22493:
22492:
22491:
22468:
22467:
22458:
22457:
22442:
22441:
22417:
22416:
22407:
22406:
22391:
22390:
22355:Fibonacci coding
22317:
22313:
22309:
22298:
22286:
22282:
22278:
22271:
22267:
22255:
22247:
22243:
22239:
22229:
22217:
22213:
22206:
22195:
22191:
22176:
22172:
22164:
22160:
22144:
22140:
22130:
22118:
22114:
22107:of a given set.
22080:
22075:
22070:
22061:
22056:
22051:
22046:
22041:
22036:
22031:
22030:
22025:
22008:
22006:
22005:
22000:
21978:
21962:
21954:
21943:
21932:
21928:
21926:
21925:
21920:
21917:
21916:
21915:
21902:
21892:
21891:
21890:
21877:
21867:
21866:
21865:
21852:
21830:
21825:
21820:
21811:
21806:
21801:
21796:
21787:
21782:
21777:
21776:
21768:
21766:
21765:
21760:
21758:
21757:
21741:
21739:
21738:
21733:
21731:
21730:
21721:
21720:
21710:
21705:
21681:
21680:
21659:
21658:
21634:
21633:
21612:
21611:
21584:
21583:
21565:
21563:
21562:
21561:
21536:
21527:
21525:
21524:
21519:
21514:
21513:
21512:
21503:
21485:
21477:
21476:
21472:
21467:
21456:
21449:
21431:
21430:
21373:
21341:Padovan sequence
21331:
21327:
21287:
21258:
21248:
21183:
21159:
21155:
21144:
21123:
21116:
21103:
21101:
21100:
21095:
21075:
21065:
21064:
21053:
21043:
21042:
21031:
21021:
21020:
21009:
20993:
20983:
20982:
20971:
20961:
20960:
20949:
20939:
20938:
20927:
20917:
20916:
20905:
20895:
20894:
20875:
20864:
20853:
20847:
20845:
20844:
20839:
20837:
20809:
20808:
20803:
20802:
20801:
20785:
20782:
20778:
20750:
20749:
20744:
20743:
20742:
20722:
20720:
20719:
20714:
20706:
20705:
20696:
20693:
20685:
20684:
20669:
20667:
20666:
20661:
20653:
20649:
20642:
20641:
20635:
20626:
20623:
20622:
20608:
20599:
20586:
20582:
20575:
20574:
20568:
20559:
20556:
20555:
20541:
20532:
20516:
20515:
20509:
20500:
20497:
20496:
20483:
20476:
20465:
20463:
20462:
20457:
20455:
20427:
20426:
20421:
20420:
20419:
20403:
20400:
20396:
20365:
20364:
20359:
20358:
20357:
20337:
20335:
20334:
20329:
20321:
20320:
20311:
20308:
20300:
20299:
20284:
20282:
20281:
20276:
20268:
20264:
20257:
20256:
20250:
20241:
20238:
20237:
20223:
20214:
20198:
20194:
20187:
20186:
20180:
20171:
20168:
20167:
20153:
20144:
20128:
20127:
20121:
20112:
20109:
20108:
20095:
20088:
20077:
20075:
20074:
20069:
20067:
20039:
20038:
20033:
20032:
20031:
20015:
20012:
20008:
19980:
19979:
19974:
19973:
19972:
19952:
19950:
19949:
19944:
19936:
19935:
19926:
19923:
19915:
19914:
19899:
19897:
19896:
19891:
19883:
19879:
19872:
19871:
19865:
19856:
19853:
19852:
19838:
19829:
19816:
19812:
19805:
19804:
19798:
19789:
19786:
19785:
19771:
19762:
19746:
19745:
19739:
19730:
19727:
19726:
19713:
19706:
19695:
19693:
19692:
19687:
19685:
19654:
19653:
19648:
19647:
19646:
19630:
19627:
19623:
19592:
19591:
19586:
19585:
19584:
19564:
19562:
19561:
19556:
19548:
19547:
19538:
19535:
19527:
19526:
19511:
19509:
19508:
19503:
19492:
19488:
19481:
19480:
19474:
19465:
19462:
19461:
19447:
19438:
19422:
19418:
19411:
19410:
19404:
19395:
19392:
19391:
19377:
19368:
19352:
19351:
19345:
19336:
19333:
19332:
19319:
19312:
19301:
19299:
19298:
19293:
19291:
19290:
19281:
19256:
19253:
19249:
19233:
19229:
19222:
19221:
19215:
19206:
19203:
19202:
19188:
19179:
19172:
19147:
19144:
19140:
19124:
19120:
19113:
19112:
19106:
19097:
19094:
19093:
19079:
19070:
19056:
19055:
19050:
19049:
19048:
19043:
19032:
19014:
18998:
18996:
18995:
18990:
18985:
18981:
18980:
18956:
18955:
18954:
18950:
18942:
18916:
18910:
18908:
18907:
18902:
18900:
18886:
18885:
18861:
18860:
18833:
18832:
18826:
18817:
18814:
18813:
18790:
18789:
18765:
18764:
18737:
18736:
18730:
18721:
18718:
18717:
18694:
18693:
18669:
18668:
18662:
18653:
18650:
18649:
18626:
18625:
18601:
18600:
18573:
18572:
18566:
18557:
18554:
18553:
18530:
18529:
18505:
18504:
18477:
18476:
18470:
18461:
18458:
18457:
18438:
18436:
18435:
18430:
18425:
18403:
18402:
18401:
18397:
18389:
18371:
18368:
18365:
18349:
18345:
18337:
18328:
18327:
18311:
18305:
18303:
18302:
18297:
18295:
18294:
18285:
18257:
18254:
18240:
18212:
18209:
18189:
18186:
18168:
18164:
18156:
18143:
18141:
18140:
18135:
18133:
18132:
18126:
18117:
18114:
18113:
18097:
18090:
18081:
18068:
18059:
18050:
18025:multiply perfect
17989:
17971:arbitrarily long
17968:
17958:
17948:is divisible by
17947:
17909:Fibonacci primes
17896:
17890:
17888:
17887:
17882:
17877:
17861:
17860:
17855:
17854:
17815:
17814:
17807:
17806:
17789:
17788:
17775:
17774:
17763:
17762:
17730:
17710:
17702:
17694:
17690:
17686:
17678:
17676:
17675:
17670:
17665:
17664:
17663:
17659:
17651:
17608:
17606:
17605:
17600:
17595:
17594:
17593:
17589:
17581:
17539:
17537:
17536:
17531:
17529:
17528:
17519:
17518:
17492:
17457:
17456:
17430:
17395:
17394:
17341:
17337:
17330:
17318:
17314:
17310:
17298:
17294:
17287:
17283:
17273:
17266:
17264:
17263:
17258:
17247:
17246:
17228:
17227:
17200:
17199:
17181:
17180:
17159:
17158:
17140:
17139:
17114:
17112:
17111:
17106:
17098:
17097:
17081:
17079:
17078:
17073:
17065:
17064:
17037:
17033:
17031:
17030:
17025:
17022:
17021:
16973:
16972:
16960:
16959:
16947:
16946:
16913:
16909:
16907:
16906:
16901:
16893:
16892:
16864:
16862:
16861:
16856:
16851:
16849:
16848:
16847:
16846:
16832:
16831:
16824:
16823:
16809:
16798:
16796:
16795:
16794:
16793:
16776:
16773:
16768:
16746:
16742:
16740:
16739:
16734:
16729:
16724:
16723:
16718:
16709:
16704:
16702:
16701:
16700:
16699:
16682:
16679:
16674:
16636:
16634:
16633:
16628:
16617:
16615:
16614:
16599:
16596:
16591:
16573:
16571:
16570:
16549:
16546:
16541:
16523:
16521:
16520:
16508:
16505:
16500:
16472:
16470:
16469:
16464:
16457:
16453:
16452:
16450:
16449:
16448:
16429:
16428:
16416:
16411:
16409:
16408:
16407:
16388:
16387:
16375:
16368:
16363:
16358:
16356:
16355:
16340:
16330:
16328:
16327:
16322:
16316:
16314:
16313:
16312:
16296:
16295:
16286:
16283:
16278:
16237:
16235:
16234:
16229:
16227:
16223:
16219:
16218:
16197:
16196:
16176:
16171:
16166:
16164:
16163:
16148:
16145:
16140:
16116:
16114:
16113:
16108:
16103:
16098:
16091:
16086:
16083:
16078:
16076:
16075:
16074:
16069:
16068:
16067:
16055:
16050:
16034:
16033:
16032:
16007:
16004:
15999:
15967:
15965:
15964:
15959:
15954:
15945:
15944:
15939:
15937:
15936:
15935:
15907:
15904:
15899:
15875:
15873:
15872:
15867:
15862:
15858:
15851:
15850:
15845:
15841:
15840:
15835:
15834:
15829:
15820:
15805:
15804:
15792:
15791:
15786:
15782:
15781:
15776:
15775:
15770:
15761:
15746:
15745:
15730:
15722:
15717:
15715:
15714:
15709:
15708:
15707:
15693:
15690:
15685:
15661:
15659:
15658:
15653:
15648:
15647:
15642:
15638:
15637:
15632:
15631:
15626:
15617:
15602:
15601:
15591:
15582:
15581:
15576:
15574:
15573:
15552:
15549:
15544:
15507:
15505:
15504:
15499:
15485:
15477:
15472:
15464:
15441:. For example,
15440:
15438:
15437:
15432:
15411:
15409:
15408:
15403:
15389:
15387:
15386:
15381:
15376:
15372:
15371:
15363:
15322:
15320:
15319:
15314:
15289:
15287:
15286:
15281:
15279:
15275:
15271:
15229:
15227:
15226:
15221:
15219:
15215:
15214:
15209:
15196:
15187:
15171:
15169:
15168:
15163:
15161:
15157:
15156:
15151:
15138:
15129:
15113:
15111:
15110:
15105:
15103:
15099:
15098:
15096:
15079:
15074:
15072:
15055:
15048:
15042:
15038:
15007:
15005:
15004:
14999:
14981:
14979:
14978:
14973:
14971:
14970:
14952:
14950:
14949:
14944:
14942:
14926:
14922:
14921:
14909:
14908:
14890:
14889:
14871:
14870:
14857:
14852:
14834:
14833:
14818:
14817:
14802:
14801:
14783:
14779:
14778:
14769:
14768:
14752:
14747:
14729:
14728:
14719:
14718:
14702:
14697:
14679:
14678:
14669:
14668:
14658:
14653:
14632:
14628:
14627:
14612:
14611:
14601:
14596:
14578:
14577:
14562:
14561:
14551:
14546:
14528:
14527:
14518:
14517:
14507:
14502:
14465:
14464:
14429:
14427:
14426:
14421:
14416:
14415:
14382:
14380:
14379:
14374:
14369:
14367:
14366:
14365:
14340:
14314:
14312:
14311:
14306:
14298:
14287:
14279:
14267:
14265:
14264:
14259:
14242:
14240:
14239:
14234:
14223:
14222:
14207:
14206:
14191:
14190:
14175:
14174:
14150:
14149:
14140:
14139:
14129:
14124:
14066:
14057:
14055:
14054:
14049:
14044:
14043:
14032:
14031:
14030:
14013:
14012:
14007:
14006:
14005:
13994:
13993:
13978:
13977:
13976:
13963:
13955:
13950:
13932:
13931:
13902:
13900:
13899:
13894:
13889:
13888:
13877:
13876:
13875:
13858:
13857:
13852:
13851:
13850:
13839:
13838:
13823:
13822:
13821:
13808:
13800:
13795:
13777:
13776:
13750:More generally,
13734:
13732:
13731:
13726:
13724:
13720:
13719:
13718:
13713:
13712:
13711:
13688:
13687:
13682:
13681:
13680:
13664:
13663:
13658:
13657:
13656:
13639:
13635:
13634:
13633:
13628:
13627:
13626:
13609:
13608:
13603:
13602:
13601:
13579:
13578:
13563:
13562:
13547:
13546:
13528:
13526:
13525:
13520:
13518:
13517:
13512:
13511:
13510:
13496:
13495:
13486:
13485:
13480:
13479:
13478:
13455:
13454:
13449:
13448:
13447:
13427:
13426:
13402:
13400:
13399:
13394:
13392:
13391:
13386:
13385:
13384:
13370:
13369:
13364:
13363:
13362:
13351:
13350:
13329:
13328:
13323:
13322:
13321:
13301:
13300:
13273:
13271:
13270:
13265:
13263:
13262:
13253:
13252:
13228:
13227:
13222:
13221:
13220:
13203:
13202:
13187:
13186:
13171:
13170:
13155:
13154:
13149:
13148:
13147:
13130:
13129:
13110:
13102:
13098:
13087:
13085:
13084:
13079:
13077:
13076:
13067:
13066:
13054:
13050:
13049:
13048:
13030:
13029:
13009:
13008:
12996:
12995:
12990:
12989:
12988:
12968:
12967:
12962:
12961:
12960:
12940:
12939:
12921:
12919:
12918:
12913:
12911:
12910:
12895:
12894:
12873:
12872:
12863:
12862:
12844:
12843:
12828:
12827:
12805:
12803:
12802:
12797:
12795:
12794:
12789:
12788:
12787:
12776:
12775:
12748:
12747:
12732:
12731:
12713:
12712:
12707:
12706:
12705:
12687:
12685:
12684:
12679:
12677:
12676:
12649:
12648:
12633:
12632:
12614:
12613:
12608:
12607:
12606:
12571:Other identities
12566:
12564:
12563:
12558:
12556:
12551:
12530:
12528:
12527:
12522:
12498:
12496:
12495:
12490:
12488:
12475:
12474:
12456:
12451:
12443:
12421:
12420:
12402:
12401:
12380:
12376:
12374:
12366:
12359:
12358:
12334:
12333:
12314:
12306:
12302:
12300:
12292:
12285:
12284:
12268:
12263:
12261:
12253:
12246:
12245:
12229:
12221:
12217:
12215:
12204:
12197:
12196:
12180:
12175:
12173:
12162:
12155:
12154:
12138:
12130:
12126:
12122:
12121:
12120:
12102:
12101:
12072:
12071:
12053:
12052:
12018:
12016:
12015:
12010:
12008:
12003:
11994:
11992:
11991:
11986:
11978:
11977:
11959:
11958:
11948:
11943:
11918:
11916:
11915:
11910:
11905:
11904:
11892:
11891:
11879:
11878:
11869:
11864:
11848:
11846:
11845:
11840:
11838:
11837:
11822:
11821:
11802:
11797:
11787:
11776:
11755:
11753:
11752:
11747:
11745:
11741:
11740:
11739:
11721:
11720:
11706:
11705:
11687:
11686:
11681:
11680:
11679:
11658:
11653:
11643:
11638:
11616:
11614:
11613:
11608:
11606:
11605:
11590:
11589:
11576:
11571:
11561:
11556:
11534:
11532:
11531:
11526:
11524:
11523:
11518:
11517:
11516:
11490:
11488:
11487:
11482:
11474:
11473:
11455:
11454:
11444:
11433:
11412:
11410:
11409:
11404:
11383:
11381:
11380:
11375:
11367:
11366:
11348:
11347:
11329:
11328:
11310:
11309:
11299:
11294:
11269:
11267:
11266:
11261:
11259:
11258:
11236:
11234:
11233:
11228:
11220:
11219:
11201:
11200:
11190:
11185:
11153:Induction proofs
11149:
11147:
11146:
11141:
11136:
11134:
11133:
11132:
11107:
11095:
11093:
11092:
11087:
11085:
11084:
11075:
11074:
11064:
11059:
11030:
11028:
11027:
11022:
10996:
10994:
10993:
10988:
10976:
10974:
10973:
10968:
10963:
10962:
10961:
10960:
10919:
10917:
10916:
10911:
10909:
10908:
10907:
10891:
10890:
10864:
10852:
10850:
10849:
10844:
10842:
10841:
10817:
10816:
10798:
10797:
10781:
10779:
10778:
10773:
10771:
10770:
10752:
10751:
10735:
10727:
10723:
10721:
10720:
10715:
10713:
10712:
10696:
10694:
10693:
10688:
10686:
10685:
10670:
10669:
10656:
10651:
10641:
10636:
10611:
10603:
10601:
10600:
10595:
10593:
10592:
10569:
10561:
10559:
10558:
10553:
10551:
10550:
10521:
10519:
10518:
10513:
10505:
10504:
10483:
10482:
10469:
10464:
10442:
10440:
10439:
10434:
10432:
10431:
10416:
10415:
10396:
10385:
10360:
10358:
10357:
10352:
10344:
10343:
10325:
10324:
10314:
10309:
10287:
10285:
10284:
10279:
10271:
10270:
10251:
10249:
10248:
10243:
10241:
10197:
10189:
10139:
10119:
10118:
10100:
10099:
10087:
10086:
10058:
10056:
10055:
10050:
9954:
9952:
9951:
9946:
9871:
9869:
9868:
9863:
9843:
9841:
9840:
9835:
9827:
9826:
9808:
9807:
9797:
9792:
9770:
9762:
9755:
9753:
9752:
9747:
9745:
9744:
9728:
9726:
9725:
9720:
9718:
9717:
9699:
9698:
9676:
9674:
9673:
9668:
9666:
9665:
9643:
9641:
9640:
9635:
9633:
9632:
9610:
9608:
9607:
9602:
9584:
9582:
9581:
9576:
9558:
9556:
9555:
9550:
9548:
9480:
9472:
9404:
9396:
9395:
9379:
9377:
9376:
9371:
9369:
9368:
9352:
9350:
9349:
9344:
9342:
9341:
9323:
9322:
9304:
9303:
9284:
9282:
9281:
9276:
9274:
9146:
9132:
9131:
9114:
9112:
9111:
9106:
9104:
9036:
9022:
9021:
9004:
9002:
9001:
8996:
8994:
8956:
8942:
8941:
8924:
8922:
8921:
8916:
8914:
8894:
8880:
8879:
8862:
8860:
8859:
8854:
8852:
8835:
8821:
8820:
8803:
8801:
8800:
8795:
8793:
8782:
8768:
8767:
8740:
8738:
8737:
8732:
8730:
8725:
8714:
8702:
8700:
8699:
8694:
8686:
8685:
8669:
8667:
8666:
8661:
8653:
8652:
8636:
8634:
8633:
8628:
8626:
8625:
8609:
8607:
8606:
8601:
8583:
8581:
8580:
8575:
8573:
8572:
8535:
8531:
8514:
8512:
8511:
8506:
8504:
8497:
8496:
8484:
8483:
8471:
8470:
8443:
8439:
8438:
8426:
8425:
8413:
8412:
8385:
8381:
8380:
8368:
8367:
8349:
8348:
8323:
8322:
8321:
8320:
8312:
8292:
8291:
8286:
8285:
8284:
8264:
8263:
8258:
8257:
8256:
8238:
8237:
8208:
8196:
8194:
8193:
8188:
8186:
8179:
8178:
8156:
8155:
8146:
8145:
8127:
8126:
8111:
8110:
8094:
8093:
8065:
8064:
8063:
8047:
8046:
8045:
8026:
8025:
8024:
8014:
8013:
8012:
7991:
7984:
7973:, the following
7972:
7963:
7951:Moreover, since
7948:
7946:
7945:
7940:
7935:
7934:
7929:
7928:
7927:
7913:
7912:
7897:
7896:
7878:
7877:
7841:
7830:
7824:
7822:
7821:
7816:
7811:
7810:
7803:
7802:
7785:
7784:
7771:
7770:
7759:
7758:
7732:
7731:
7726:
7725:
7680:
7659:
7651:
7647:
7617:
7607:
7605:
7604:
7599:
7594:
7592:
7590:
7589:
7587:
7585:
7584:
7582:
7580:
7564:
7562:
7552:
7538:
7536:
7526:
7512:
7510:
7500:
7477:
7458:
7450:
7448:
7447:
7442:
7437:
7435:
7433:
7428:
7421:
7419:
7418:
7417:
7393:
7392:
7377:
7372:
7371:
7353:
7351:
7350:
7345:
7343:
7336:
7335:
7334:
7321:
7314:
7313:
7291:
7290:
7265:
7259:
7255:
7253:
7252:
7245:
7244:
7209:
7208:
7191:
7190:
7171:
7170:
7136:
7132:
7131:
7130:
7124:
7123:
7114:
7113:
7103:
7096:
7095:
7083:
7082:
7075:
7074:
7039:
7038:
7012:
7008:
7007:
7006:
7000:
6999:
6990:
6989:
6979:
6972:
6971:
6959:
6958:
6940:
6936:
6935:
6934:
6928:
6927:
6918:
6917:
6907:
6900:
6899:
6883:
6882:
6881:
6875:
6874:
6865:
6864:
6848:
6828:
6822:
6820:
6819:
6814:
6809:
6808:
6789:
6788:
6750:
6749:
6741:
6740:
6683:
6681:
6680:
6675:
6673:
6666:
6665:
6653:
6652:
6633:
6632:
6616:
6615:
6570:
6558:
6556:
6555:
6550:
6545:
6544:
6538:
6534:
6532:
6530:
6520:
6518:
6517:
6512:
6498:
6491:
6489:
6487:
6482:
6475:
6473:
6463:
6457:
6456:
6450:
6446:
6444:
6442:
6432:
6430:
6429:
6424:
6410:
6403:
6401:
6399:
6394:
6387:
6385:
6375:
6370:
6369:
6349:
6343:
6341:
6340:
6335:
6333:
6326:
6325:
6324:
6315:
6314:
6313:
6296:
6289:
6288:
6282:
6278:
6276:
6274:
6264:
6262:
6261:
6256:
6242:
6235:
6233:
6231:
6226:
6219:
6217:
6207:
6200:
6199:
6198:
6185:
6178:
6177:
6171:
6167:
6165:
6163:
6153:
6151:
6150:
6145:
6131:
6124:
6122:
6120:
6115:
6108:
6106:
6096:
6088:
6084:
6083:
6075:
6072:
6071:
6050:
6044:
6040:
6035:
6034:
6026:
6023:
6022:
6013:
6007:
6003:
5995:
5991:
5990:
5982:
5979:
5978:
5969:
5963:
5959:
5954:
5953:
5945:
5942:
5941:
5932:
5926:
5922:
5913:
5912:
5907:
5906:
5898:
5883:
5877:
5875:
5874:
5869:
5864:
5863:
5855:
5852:
5846:
5842:
5837:
5836:
5828:
5825:
5819:
5815:
5810:
5809:
5808:
5795:
5785:
5784:
5779:
5778:
5770:
5757:
5755:
5754:
5749:
5744:
5743:
5742:
5733:
5732:
5731:
5714:
5704:
5703:
5695:
5688:
5687:
5686:
5673:
5663:
5662:
5654:
5641:
5639:
5638:
5633:
5631:
5630:
5621:
5620:
5615:
5607:
5606:
5600:
5591:
5585:
5584:
5556:
5554:
5553:
5548:
5546:
5545:
5536:
5535:
5530:
5522:
5521:
5515:
5506:
5490:
5477:
5475:
5474:
5469:
5467:
5466:
5461:
5460:
5452:
5448:
5447:
5442:
5433:
5432:
5427:
5426:
5418:
5405:
5403:
5402:
5397:
5392:
5391:
5386:
5385:
5377:
5373:
5365:
5364:
5353:
5352:
5344:
5333:
5331:
5330:
5325:
5323:
5322:
5321:
5315:
5314:
5305:
5304:
5288:
5281:
5280:
5242:
5241:
5240:
5234:
5233:
5218:
5217:
5201:
5170:
5168:
5167:
5162:
5156:
5155:
5146:
5145:
5124:
5123:
5108:
5107:
5083:
5082:
5077:
5076:
5075:
5051:
5049:
5048:
5043:
5038:
5030:
5029:
5008:
5007:
4995:
4990:
4988:
4987:
4972:
4971:
4950:
4949:
4937:
4936:
4918:
4916:
4915:
4910:
4905:
4900:
4899:
4897:
4896:
4872:
4871:
4866:
4865:
4864:
4850:
4845:
4840:
4838:
4837:
4827:
4822:
4821:
4799:
4797:
4796:
4791:
4789:
4788:
4769:and solved as a
4768:
4766:
4765:
4760:
4758:
4757:
4748:
4743:
4734:
4732:
4731:
4726:
4724:
4719:
4717:
4709:
4708:
4696:
4695:
4674:
4673:
4658:
4657:
4637:
4635:
4634:
4629:
4621:
4620:
4601:
4599:
4598:
4593:
4585:
4584:
4566:at least one of
4561:
4550:
4548:
4547:
4542:
4537:
4536:
4518:
4517:
4499:
4498:
4471:
4462:
4460:
4459:
4454:
4449:
4448:
4427:
4426:
4414:
4413:
4397:
4395:
4394:
4389:
4375:
4374:
4358:
4356:
4355:
4350:
4345:
4321:
4319:
4318:
4313:
4308:
4307:
4292:
4291:
4273:
4272:
4254:
4253:
4235:
4234:
4216:
4215:
4182:
4181:
4166:
4165:
4147:
4146:
4137:
4136:
4118:
4117:
4096:
4095:
4080:
4079:
4057:
4042:
4040:
4039:
4034:
4029:
4028:
4007:
4006:
3994:
3993:
3969:
3967:
3966:
3961:
3949:
3947:
3946:
3941:
3939:
3938:
3920:
3918:
3917:
3912:
3895:
3894:
3858:
3856:
3855:
3850:
3838:
3836:
3835:
3830:
3828:
3827:
3815:
3813:
3812:
3803:
3802:
3787:
3784:
3755:
3753:
3752:
3747:
3742:
3737:
3736:
3721:
3720:
3704:
3702:
3701:
3696:
3694:
3693:
3677:
3675:
3674:
3669:
3667:
3666:
3648:
3646:
3645:
3640:
3629:
3627:
3626:
3617:
3616:
3601:
3598:
3577:
3575:
3574:
3569:
3540:
3538:
3537:
3532:
3527:
3525:
3514:
3500:
3489:
3488:
3469:
3458:
3449:decimal digits.
3448:
3444:
3437:
3435:
3434:
3429:
3411:
3410:
3391:
3380:
3378:
3377:
3372:
3370:
3365:
3363:
3358:
3357:
3322:
3320:
3319:
3314:
3291:
3290:
3274:
3272:
3271:
3266:
3233:
3231:
3230:
3225:
3211:
3210:
3201:
3184:
3183:
3156:
3121:
3120:
3104:
3102:
3101:
3096:
3080:
3076:
3072:
3065:
3048:
3043:
3038:
3037:
3011:
3010:
3009:
2974:
2964:
2962:
2961:
2956:
2943:
2939:
2935:
2931:
2926:
2921:
2920:
2884:
2880:
2873:
2866:
2860:
2858:
2857:
2852:
2839:
2835:
2831:
2825:
2824:
2823:
2814:
2805:
2804:
2784:
2782:
2781:
2776:
2774:
2768:
2767:
2766:
2757:
2744:
2733:
2726:
2724:
2723:
2718:
2716:
2708:
2703:
2699:
2693:
2692:
2691:
2682:
2662:
2660:
2659:
2654:
2652:
2645:
2639:
2638:
2637:
2636:
2621:
2620:
2610:
2591:
2585:
2584:
2580:
2579:
2567:
2566:
2556:
2529:
2527:
2526:
2521:
2519:
2518:
2503:
2502:
2487:
2486:
2468:
2459:
2445:
2443:
2442:
2437:
2416:
2410:
2406:
2401:
2399:
2385:
2366:
2364:
2363:
2358:
2356:
2353:
2284:
2280:
2276:
2265:
2255:
2245:
2241:
2235:
2233:
2232:
2227:
2225:
2218:
2217:
2199:
2198:
2177:
2173:
2172:
2151:
2150:
2129:
2128:
2107:
2106:
2082:
2075:
2074:
2056:
2055:
2028:
2027:
2009:
2008:
1981:
1977:
1976:
1961:
1960:
1941:
1940:
1916:
1914:
1913:
1908:
1906:
1905:
1890:
1889:
1874:
1873:
1855:
1851:
1845:
1843:
1842:
1837:
1835:
1828:
1827:
1809:
1808:
1786:
1785:
1769:
1768:
1750:
1749:
1727:
1726:
1704:
1700:
1696:
1694:
1693:
1688:
1683:
1682:
1664:
1663:
1645:
1644:
1628:
1626:
1625:
1620:
1606:
1605:
1589:
1585:
1579:
1577:
1576:
1571:
1566:
1564:
1550:
1549:
1548:
1524:
1523:
1513:
1508:
1502:
1501:
1500:
1499:
1475:
1474:
1464:
1459:
1458:
1440:
1438:
1437:
1432:
1430:
1429:
1399:
1397:
1396:
1391:
1370:
1362:
1342:
1337:
1336:
1331:
1322:
1301:
1291:
1289:
1288:
1283:
1268:
1263:
1262:
1257:
1248:
1229:
1227:
1226:
1221:
1216:
1210:
1209:
1208:
1207:
1195:
1194:
1184:
1179:
1177:
1166:
1165:
1164:
1152:
1151:
1141:
1136:
1135:
1116:Daniel Bernoulli
1062:the end of the n
1041:
1037:
1033:
1026:
1019:
977:
959:
906:
894:
883:
871:
863:
852:
840:
835:Sanskrit prosody
824:
815:
806:
708:
698:
688:
678:
668:
658:
648:
638:
628:
618:
608:
598:
588:
578:
568:
558:
548:
538:
528:
518:
508:
507:
502:
490:
483:
481:
480:
475:
473:
472:
454:
453:
435:
434:
418:
416:
415:
410:
399:
398:
386:
385:
369:
367:
366:
361:
353:
352:
333:
326:
324:
323:
318:
316:
315:
297:
296:
278:
277:
261:
259:
258:
253:
242:
241:
222:
221:
166:
162:
158:
95:
51:
50:
47:
28922:
28921:
28917:
28916:
28915:
28913:
28912:
28911:
28892:
28891:
28890:
28885:
28845:
28824:
28797:
28792:
28762:
28757:
28739:
28696:
28645:Kinds of series
28636:
28575:
28542:Explicit series
28533:
28507:
28469:
28455:Cauchy sequence
28443:
28430:
28384:Figurate number
28361:
28355:
28346:Powers of three
28290:
28281:
28251:
28246:
28237:
28170:Kepler triangle
28136:
28131:
28101:
28096:
28074:
28070:Strobogrammatic
28061:
28043:
28025:
28007:
27989:
27971:
27953:
27935:
27912:
27891:
27875:
27834:Divisor-related
27829:
27789:
27740:
27710:
27647:
27631:
27610:
27577:
27550:
27538:
27520:
27432:
27431:related numbers
27405:
27382:
27349:
27340:Perfect totient
27306:
27283:
27214:Highly abundant
27156:
27135:
27067:
27050:
27022:
27005:
26991:Stirling second
26897:
26874:
26835:
26817:
26774:
26723:
26660:
26621:Centered square
26589:
26572:
26534:
26519:
26486:
26471:
26423:
26422:defined numbers
26405:
26372:
26357:
26328:Double Mersenne
26314:
26295:
26217:
26203:
26201:natural numbers
26197:
26142:
26105:
26101:
26092:
26062:
26034:
26016:
25988:
25966:
25947:
25927:
25922:
25921:
25913:
25909:
25901:
25897:
25857:
25853:
25817:
25813:
25802:
25801:
25797:
25789:
25785:
25777:
25775:
25771:
25760:
25754:
25747:
25742:
25738:
25722:
25718:
25717:
25713:
25707:
25676:
25672:
25664:
25660:
25648:(3â4): 179â89,
25638:
25634:
25628:
25618:Springer-Verlag
25610:
25606:
25593:
25591:
25589:
25573:
25569:
25561:
25557:
25549:
25545:
25533:
25525:
25521:
25508:
25507:
25503:
25490:
25486:
25480:
25466:
25462:
25454:
25431:
25425:
25421:
25413:
25411:
25402:
25398:
25391:Multimedia Wiki
25387:
25386:
25382:
25373:
25372:
25368:
25357:
25353:
25328:
25324:
25318:
25304:Knuth, Donald E
25301:
25297:
25278:
25262:
25258:
25236:
25232:
25209:
25205:
25199:
25185:
25181:
25173:
25169:
25144:
25140:
25135:
25131:
25110:
25106:
25092:10.2307/2325076
25076:
25072:
25061:
25050:
25041:
25040:
25036:
25026:
25015:
25006:
25005:
25001:
24993:
24989:
24981:
24977:
24969:
24965:
24959:
24942:
24938:
24917:
24913:
24900:
24896:
24862:
24858:
24832:
24828:
24782:
24778:
24762:
24756:
24752:
24704:
24700:
24685:
24681:
24651:
24647:
24641:
24627:
24623:
24615:
24613:
24609:
24595:
24576:
24570:Diaconis, Persi
24567:
24563:
24542:
24538:
24530:
24526:
24465:
24453:
24449:
24447:
24444:
24443:
24440:
24436:
24428:
24426:
24413:
24409:
24395:
24391:
24385:
24367:
24363:
24339:
24335:
24318:
24310:
24306:
24294:
24290:
24268:10.2307/3618079
24262:(486): 521â25,
24252:
24248:
24227:
24218:
24212:
24196:
24192:
24186:
24172:
24168:
24160:
24156:
24149:
24138:
24134:
24125:
24123:
24121:nrich.maths.org
24115:
24114:
24110:
24101:
24099:
24085:
24079:
24075:
24066:
24065:
24061:
24055:
24041:
24037:
24016:
24012:
23991:
23987:
23979:
23975:
23967:
23963:
23957:
23935:
23931:
23921:
23905:
23901:
23890:
23876:Gardner, Martin
23873:
23869:
23852:
23848:
23839:
23837:
23825:
23821:
23815:
23801:
23797:
23788:
23786:
23775:
23774:
23770:
23762:
23758:
23749:
23747:
23737:
23731:
23727:
23719:
23715:
23704:
23700:
23670:
23666:
23648:
23644:
23636:
23629:
23618:
23599:
23595:
23587:
23583:
23575:
23571:
23563:
23559:
23551:
23547:
23535:
23530:
23511:
23504:
23480:
23473:
23467:
23451:
23442:
23421:
23414:
23409:
23404:
23403:
23394:
23390:
23385:
23380:
23344:
23307:resistor ladder
23299:
23263:
23259:
23252:
23248:
23236:
23202:
23198:
23196:
23193:
23192:
23190:
23165:
23161:
23159:
23156:
23155:
23153:
23128:
23124:
23122:
23119:
23118:
23116:
23091:
23087:
23085:
23082:
23081:
23079:
23056:
23052:
23050:
23047:
23046:
23023:
23014:
23011:
23002:
22999:
22991:
22960:
22956:
22955:for some index
22941:
22937:
22918:
22907:Fermat's spiral
22902:
22898:
22878:
22852:
22848:
22840:
22838:
22830:
22827:
22826:
22816:
22795:
22763:Alexander Braun
22708:
22702:
22696:
22635:data structure.
22593:
22566:
22563:Balance factors
22555:
22494:
22478:
22474:
22473:
22472:
22463:
22459:
22447:
22443:
22431:
22427:
22412:
22408:
22396:
22392:
22386:
22382:
22377:
22374:
22373:
22328:, which led to
22315:
22311:
22307:
22300:
22297:
22288:
22284:
22280:
22276:
22269:
22266:
22257:
22249:
22245:
22241:
22237:
22231:
22228:
22219:
22215:
22211:
22201:
22193:
22178:
22174:
22166:
22162:
22159:
22150:
22142:
22138:
22132:
22129:
22120:
22116:
22112:
22078:
22073:
22068:
22059:
22054:
22049:
22044:
22039:
22034:
22023:
22016:
22010:
21988:
21985:
21984:
21977:
21968:
21960:
21952:
21934:
21930:
21911:
21898:
21897:
21896:
21886:
21873:
21872:
21871:
21861:
21848:
21847:
21846:
21843:
21840:
21839:
21828:
21823:
21818:
21809:
21804:
21799:
21794:
21785:
21780:
21753:
21749:
21747:
21744:
21743:
21726:
21722:
21716:
21712:
21706:
21695:
21676:
21672:
21648:
21644:
21629:
21625:
21607:
21603:
21579:
21575:
21557:
21553:
21540:
21535:
21533:
21530:
21529:
21508:
21487:
21481:
21480:
21479:
21457:
21455:
21451:
21450:
21439:
21426:
21422:
21420:
21417:
21416:
21393:
21388:
21348:
21329:
21326:
21316:
21305:
21300:
21286:
21276:
21265:
21260:
21256:
21250:
21246:
21240:
21200:
21194:
21192:Generalizations
21186:cycle detection
21181:
21174:modular integer
21157:
21150:
21142:
21139:
21133:
21118:
21107:
21069:
21060:
21056:
21038:
21034:
21016:
21012:
20987:
20978:
20974:
20956:
20952:
20934:
20930:
20912:
20908:
20890:
20886:
20884:
20881:
20880:
20874:
20866:
20863:
20855:
20851:
20822:
20804:
20797:
20793:
20792:
20791:
20783: and
20781:
20763:
20745:
20738:
20734:
20733:
20732:
20727:
20724:
20723:
20701:
20697:
20694: and
20692:
20680:
20676:
20674:
20671:
20670:
20637:
20636:
20624:
20618:
20617:
20613:
20609:
20597:
20570:
20569:
20557:
20551:
20550:
20546:
20542:
20530:
20511:
20510:
20498:
20492:
20491:
20489:
20486:
20485:
20478:
20477:, in this case
20471:
20440:
20422:
20415:
20411:
20410:
20409:
20401: and
20399:
20381:
20360:
20353:
20349:
20348:
20347:
20342:
20339:
20338:
20316:
20312:
20309: and
20307:
20295:
20291:
20289:
20286:
20285:
20252:
20251:
20239:
20233:
20232:
20228:
20224:
20212:
20182:
20181:
20169:
20163:
20162:
20158:
20154:
20142:
20123:
20122:
20110:
20104:
20103:
20101:
20098:
20097:
20090:
20089:, in this case
20083:
20052:
20034:
20027:
20023:
20022:
20021:
20013: and
20011:
19993:
19975:
19968:
19964:
19963:
19962:
19957:
19954:
19953:
19931:
19927:
19924: and
19922:
19910:
19906:
19904:
19901:
19900:
19867:
19866:
19854:
19848:
19847:
19843:
19839:
19827:
19800:
19799:
19787:
19781:
19780:
19776:
19772:
19760:
19741:
19740:
19728:
19722:
19721:
19719:
19716:
19715:
19708:
19707:, in this case
19701:
19670:
19649:
19642:
19638:
19637:
19636:
19628: and
19626:
19608:
19587:
19580:
19576:
19575:
19574:
19569:
19566:
19565:
19543:
19539:
19536: and
19534:
19522:
19518:
19516:
19513:
19512:
19476:
19475:
19463:
19457:
19456:
19452:
19448:
19436:
19406:
19405:
19393:
19387:
19386:
19382:
19378:
19366:
19347:
19346:
19334:
19328:
19327:
19325:
19322:
19321:
19314:
19313:, in this case
19307:
19286:
19285:
19266:
19252:
19250:
19234:
19217:
19216:
19204:
19198:
19197:
19193:
19189:
19177:
19174:
19173:
19157:
19143:
19141:
19125:
19108:
19107:
19095:
19089:
19088:
19084:
19080:
19068:
19061:
19060:
19051:
19033:
19031:
19027:
19026:
19025:
19020:
19017:
19016:
19009:
18976:
18972:
18963:
18941:
18937:
18930:
18926:
18924:
18921:
18920:
18914:
18898:
18897:
18887:
18881:
18877:
18875:
18862:
18856:
18852:
18850:
18834:
18828:
18827:
18815:
18809:
18808:
18805:
18804:
18791:
18785:
18781:
18779:
18766:
18760:
18756:
18754:
18738:
18732:
18731:
18719:
18713:
18712:
18709:
18708:
18695:
18689:
18685:
18683:
18670:
18664:
18663:
18651:
18645:
18644:
18641:
18640:
18627:
18621:
18617:
18615:
18602:
18596:
18592:
18590:
18574:
18568:
18567:
18555:
18549:
18548:
18545:
18544:
18531:
18525:
18521:
18519:
18506:
18500:
18496:
18494:
18478:
18472:
18471:
18459:
18453:
18452:
18448:
18446:
18443:
18442:
18410:
18388:
18384:
18377:
18373:
18367:
18350:
18336:
18332:
18323:
18319:
18317:
18314:
18313:
18309:
18290:
18289:
18270:
18253:
18251:
18242:
18241:
18225:
18208:
18206:
18200:
18199:
18185:
18183:
18173:
18172:
18155:
18151:
18149:
18146:
18145:
18128:
18127:
18115:
18109:
18108:
18106:
18103:
18102:
18100:Legendre symbol
18095:
18089:
18083:
18080:
18074:
18067:
18061:
18058:
18052:
18049:
18042:
18036:
18033:
17987:
17981:
17966:
17960:
17957:
17949:
17946:
17938:
17921:Fibonacci prime
17917:
17915:Fibonacci prime
17911:
17901:, which can be
17892:
17862:
17856:
17849:
17848:
17843:
17837:
17836:
17831:
17821:
17820:
17819:
17809:
17808:
17796:
17792:
17790:
17784:
17780:
17777:
17776:
17770:
17766:
17764:
17752:
17748:
17741:
17740:
17738:
17735:
17734:
17724:
17716:
17708:
17700:
17692:
17688:
17684:
17650:
17646:
17637:
17633:
17625:
17622:
17621:
17614:
17580:
17576:
17567:
17563:
17555:
17552:
17551:
17548:Legendre symbol
17524:
17523:
17508:
17504:
17493:
17477:
17462:
17461:
17446:
17442:
17431:
17415:
17400:
17399:
17390:
17386:
17375:
17359:
17358:
17356:
17353:
17352:
17347:
17339:
17332:
17329:
17320:
17316:
17312:
17309:
17300:
17296:
17292:
17285:
17281:
17271:
17236:
17232:
17217:
17213:
17189:
17185:
17176:
17172:
17148:
17144:
17135:
17131:
17123:
17120:
17119:
17093:
17089:
17087:
17084:
17083:
17060:
17056:
17054:
17051:
17050:
17035:
16990:
16986:
16968:
16964:
16955:
16951:
16942:
16938:
16930:
16927:
16926:
16919:
16911:
16888:
16884:
16882:
16879:
16878:
16875:
16870:
16842:
16838:
16837:
16833:
16819:
16815:
16814:
16810:
16808:
16789:
16785:
16784:
16780:
16775:
16769:
16758:
16752:
16749:
16748:
16744:
16717:
16710:
16708:
16695:
16691:
16690:
16686:
16681:
16675:
16664:
16658:
16655:
16654:
16651:Millin's series
16607:
16603:
16598:
16592:
16581:
16557:
16553:
16548:
16542:
16531:
16516:
16512:
16507:
16501:
16490:
16484:
16481:
16480:
16441:
16437:
16430:
16421:
16417:
16415:
16400:
16396:
16389:
16380:
16376:
16374:
16373:
16369:
16362:
16348:
16344:
16339:
16336:
16333:
16332:
16308:
16304:
16297:
16291:
16287:
16285:
16279:
16268:
16246:
16243:
16242:
16214:
16210:
16192:
16188:
16181:
16177:
16170:
16156:
16152:
16147:
16141:
16130:
16124:
16121:
16120:
16085:
16084:
16082:
16070:
16063:
16059:
16058:
16057:
16051:
16040:
16035:
16022:
16018:
16008:
16006:
16000:
15989:
15983:
15980:
15979:
15970:and there is a
15943:
15922:
15918:
15911:
15906:
15900:
15889:
15883:
15880:
15879:
15846:
15828:
15821:
15819:
15812:
15808:
15807:
15800:
15796:
15787:
15769:
15762:
15760:
15753:
15749:
15748:
15741:
15737:
15736:
15732:
15721:
15710:
15703:
15699:
15698:
15697:
15692:
15686:
15675:
15669:
15666:
15665:
15643:
15625:
15618:
15616:
15609:
15605:
15604:
15597:
15593:
15580:
15560:
15556:
15551:
15545:
15534:
15528:
15525:
15524:
15521:theta functions
15513:
15511:Reciprocal sums
15476:
15463:
15446:
15443:
15442:
15417:
15414:
15413:
15397:
15394:
15393:
15362:
15358:
15354:
15333:
15330:
15329:
15299:
15296:
15295:
15267:
15260:
15256:
15242:
15239:
15238:
15208:
15201:
15197:
15185:
15177:
15174:
15173:
15150:
15143:
15139:
15127:
15119:
15116:
15115:
15083:
15078:
15059:
15054:
15053:
15049:
15037:
15020:
15017:
15016:
14987:
14984:
14983:
14966:
14962:
14960:
14957:
14956:
14940:
14939:
14924:
14923:
14917:
14913:
14898:
14894:
14879:
14875:
14866:
14862:
14853:
14842:
14829:
14825:
14813:
14809:
14797:
14793:
14781:
14780:
14774:
14770:
14758:
14754:
14748:
14737:
14724:
14720:
14708:
14704:
14698:
14687:
14674:
14670:
14664:
14660:
14654:
14643:
14630:
14629:
14617:
14613:
14607:
14603:
14597:
14586:
14567:
14563:
14557:
14553:
14547:
14536:
14523:
14519:
14513:
14509:
14503:
14492:
14481:
14460:
14456:
14437:
14435:
14432:
14431:
14411:
14407:
14390:
14387:
14386:
14361:
14357:
14344:
14339:
14322:
14319:
14318:
14294:
14283:
14275:
14273:
14270:
14269:
14253:
14250:
14249:
14218:
14214:
14202:
14198:
14186:
14182:
14170:
14166:
14145:
14141:
14135:
14131:
14125:
14114:
14093:
14090:
14089:
14077:
14061:
14033:
14020:
14016:
14015:
14014:
14008:
14001:
13997:
13996:
13995:
13983:
13979:
13972:
13959:
13958:
13957:
13951:
13940:
13918:
13914:
13912:
13909:
13908:
13878:
13865:
13861:
13860:
13859:
13853:
13846:
13842:
13841:
13840:
13828:
13824:
13817:
13804:
13803:
13802:
13796:
13785:
13763:
13759:
13757:
13754:
13753:
13714:
13701:
13697:
13696:
13695:
13683:
13676:
13672:
13671:
13670:
13669:
13665:
13659:
13652:
13648:
13647:
13646:
13629:
13622:
13618:
13617:
13616:
13604:
13591:
13587:
13586:
13585:
13584:
13580:
13568:
13564:
13558:
13554:
13539:
13535:
13533:
13530:
13529:
13513:
13506:
13502:
13501:
13500:
13491:
13487:
13481:
13468:
13464:
13463:
13462:
13450:
13437:
13433:
13432:
13431:
13413:
13409:
13407:
13404:
13403:
13387:
13380:
13376:
13375:
13374:
13365:
13358:
13354:
13353:
13352:
13340:
13336:
13324:
13311:
13307:
13306:
13305:
13287:
13283:
13281:
13278:
13277:
13258:
13254:
13248:
13244:
13223:
13216:
13212:
13211:
13210:
13192:
13188:
13176:
13172:
13166:
13162:
13150:
13143:
13139:
13138:
13137:
13122:
13118:
13116:
13113:
13112:
13108:
13100:
13097:
13089:
13072:
13068:
13062:
13058:
13038:
13034:
13019:
13015:
13014:
13010:
13004:
13000:
12991:
12978:
12974:
12973:
12972:
12963:
12950:
12946:
12945:
12944:
12932:
12928:
12926:
12923:
12922:
12900:
12896:
12890:
12886:
12868:
12864:
12852:
12848:
12833:
12829:
12823:
12819:
12817:
12814:
12813:
12811:
12790:
12783:
12779:
12778:
12777:
12765:
12761:
12737:
12733:
12721:
12717:
12708:
12701:
12697:
12696:
12695:
12693:
12690:
12689:
12666:
12662:
12638:
12634:
12622:
12618:
12609:
12602:
12598:
12597:
12596:
12594:
12591:
12590:
12587:
12581:
12573:
12550:
12536:
12533:
12532:
12504:
12501:
12500:
12486:
12485:
12464:
12460:
12450:
12441:
12440:
12410:
12406:
12391:
12387:
12378:
12377:
12367:
12348:
12344:
12323:
12319:
12315:
12313:
12304:
12303:
12293:
12274:
12270:
12269:
12267:
12254:
12235:
12231:
12230:
12228:
12219:
12218:
12205:
12186:
12182:
12181:
12179:
12163:
12144:
12140:
12139:
12137:
12128:
12127:
12116:
12112:
12097:
12093:
12080:
12076:
12067:
12063:
12048:
12044:
12036:
12026:
12024:
12021:
12020:
12002:
12000:
11997:
11996:
11967:
11963:
11954:
11950:
11944:
11933:
11927:
11924:
11923:
11900:
11896:
11887:
11883:
11874:
11870:
11863:
11861:
11858:
11857:
11854:
11827:
11823:
11811:
11807:
11798:
11793:
11777:
11766:
11760:
11757:
11756:
11729:
11725:
11716:
11712:
11711:
11707:
11695:
11691:
11682:
11669:
11665:
11664:
11663:
11654:
11649:
11639:
11628:
11622:
11619:
11618:
11595:
11591:
11585:
11581:
11572:
11567:
11557:
11546:
11540:
11537:
11536:
11519:
11506:
11502:
11501:
11500:
11498:
11495:
11494:
11493:Similarly, add
11463:
11459:
11450:
11446:
11434:
11423:
11417:
11414:
11413:
11392:
11389:
11388:
11356:
11352:
11337:
11333:
11318:
11314:
11305:
11301:
11295:
11284:
11278:
11275:
11274:
11248:
11244:
11242:
11239:
11238:
11209:
11205:
11196:
11192:
11186:
11175:
11169:
11166:
11165:
11155:
11128:
11124:
11111:
11106:
11104:
11101:
11100:
11080:
11076:
11070:
11066:
11060:
11049:
11043:
11040:
11039:
11010:
11007:
11006:
10982:
10979:
10978:
10956:
10952:
10945:
10944:
10933:
10930:
10929:
10922:symbolic method
10903:
10896:
10892:
10886:
10882:
10877:
10874:
10873:
10870:
10868:Symbolic method
10837:
10833:
10806:
10802:
10793:
10789:
10787:
10784:
10783:
10760:
10756:
10747:
10743:
10741:
10738:
10737:
10729:
10725:
10708:
10704:
10702:
10699:
10698:
10675:
10671:
10665:
10661:
10652:
10647:
10637:
10626:
10620:
10617:
10616:
10605:
10585:
10581:
10579:
10576:
10575:
10563:
10537:
10533:
10531:
10528:
10527:
10491:
10487:
10475:
10471:
10465:
10454:
10448:
10445:
10444:
10424:
10420:
10402:
10398:
10386:
10375:
10369:
10366:
10365:
10333:
10329:
10320:
10316:
10310:
10299:
10293:
10290:
10289:
10266:
10262:
10260:
10257:
10256:
10237:
10193:
10185:
10135:
10108:
10104:
10095:
10091:
10076:
10072:
10070:
10067:
10066:
9960:
9957:
9956:
9877:
9874:
9873:
9851:
9848:
9847:
9816:
9812:
9803:
9799:
9793:
9782:
9776:
9773:
9772:
9764:
9760:
9740:
9736:
9734:
9731:
9730:
9707:
9703:
9688:
9684:
9682:
9679:
9678:
9655:
9651:
9649:
9646:
9645:
9622:
9618:
9616:
9613:
9612:
9590:
9587:
9586:
9564:
9561:
9560:
9544:
9476:
9468:
9400:
9391:
9387:
9385:
9382:
9381:
9364:
9360:
9358:
9355:
9354:
9331:
9327:
9312:
9308:
9299:
9295:
9293:
9290:
9289:
9270:
9142:
9127:
9123:
9121:
9118:
9117:
9100:
9032:
9017:
9013:
9011:
9008:
9007:
8990:
8952:
8937:
8933:
8931:
8928:
8927:
8910:
8890:
8875:
8871:
8869:
8866:
8865:
8848:
8831:
8816:
8812:
8810:
8807:
8806:
8789:
8778:
8763:
8759:
8757:
8754:
8753:
8726:
8715:
8710:
8708:
8705:
8704:
8681:
8677:
8675:
8672:
8671:
8648:
8644:
8642:
8639:
8638:
8621:
8617:
8615:
8612:
8611:
8589:
8586:
8585:
8568:
8564:
8562:
8559:
8558:
8551:
8546:
8533:
8522:
8502:
8501:
8492:
8488:
8479:
8475:
8460:
8456:
8441:
8440:
8434:
8430:
8421:
8417:
8402:
8398:
8383:
8382:
8376:
8372:
8357:
8353:
8338:
8334:
8324:
8311:
8309:
8308:
8301:
8297:
8294:
8293:
8287:
8274:
8270:
8269:
8268:
8259:
8252:
8248:
8247:
8246:
8239:
8224:
8220:
8216:
8214:
8211:
8210:
8200:
8184:
8183:
8168:
8164:
8157:
8151:
8147:
8135:
8131:
8116:
8112:
8106:
8102:
8099:
8098:
8077:
8073:
8066:
8053:
8049:
8048:
8035:
8031:
8030:
8020:
8016:
8015:
8008:
8004:
8003:
7999:
7997:
7994:
7993:
7986:
7982:
7968:
7952:
7930:
7923:
7919:
7918:
7917:
7902:
7898:
7886:
7882:
7873:
7869:
7858:
7855:
7854:
7832:
7828:
7805:
7804:
7792:
7788:
7786:
7780:
7776:
7773:
7772:
7766:
7762:
7760:
7748:
7744:
7737:
7736:
7727:
7720:
7719:
7714:
7708:
7707:
7702:
7692:
7691:
7690:
7688:
7685:
7684:
7679:
7670:
7661:
7653:
7649:
7646:
7637:
7627:
7619:
7615:
7570:
7565:
7558:
7553:
7551:
7544:
7539:
7532:
7527:
7525:
7518:
7513:
7506:
7501:
7499:
7485:
7482:
7481:
7475:
7454:
7427:
7422:
7410:
7406:
7388:
7384:
7383:
7378:
7376:
7367:
7363:
7361:
7358:
7357:
7341:
7340:
7330:
7317:
7316:
7315:
7308:
7307:
7302:
7293:
7292:
7283:
7279:
7277:
7267:
7266:
7254:
7247:
7246:
7237:
7233:
7222:
7216:
7215:
7210:
7204:
7200:
7193:
7192:
7185:
7184:
7179:
7173:
7172:
7163:
7159:
7154:
7144:
7143:
7134:
7133:
7126:
7119:
7115:
7109:
7105:
7099:
7098:
7097:
7088:
7084:
7077:
7076:
7067:
7063:
7052:
7046:
7045:
7040:
7034:
7030:
7023:
7022:
7010:
7009:
7002:
6995:
6991:
6985:
6981:
6975:
6974:
6973:
6964:
6960:
6954:
6950:
6938:
6937:
6930:
6923:
6919:
6913:
6909:
6903:
6902:
6901:
6895:
6891:
6884:
6877:
6870:
6866:
6854:
6850:
6844:
6843:
6842:
6838:
6836:
6833:
6832:
6826:
6803:
6802:
6797:
6791:
6790:
6781:
6777:
6772:
6762:
6761:
6744:
6743:
6733:
6729:
6724:
6718:
6717:
6712:
6702:
6701:
6693:
6690:
6689:
6671:
6670:
6658:
6654:
6648:
6644:
6634:
6628:
6624:
6621:
6620:
6608:
6604:
6591:
6584:
6582:
6579:
6578:
6566:
6563:diagonalization
6539:
6526:
6521:
6511:
6504:
6499:
6497:
6493:
6492:
6481:
6476:
6469:
6464:
6462:
6451:
6438:
6433:
6423:
6416:
6411:
6409:
6405:
6404:
6393:
6388:
6381:
6376:
6374:
6365:
6361:
6359:
6356:
6355:
6347:
6346:From this, the
6331:
6330:
6320:
6306:
6302:
6298:
6292:
6291:
6290:
6283:
6270:
6265:
6255:
6248:
6243:
6241:
6237:
6236:
6225:
6220:
6213:
6208:
6206:
6194:
6181:
6180:
6179:
6172:
6159:
6154:
6144:
6137:
6132:
6130:
6126:
6125:
6114:
6109:
6102:
6097:
6095:
6086:
6085:
6074:
6073:
6064:
6060:
6039:
6025:
6024:
6018:
6014:
6002:
5993:
5992:
5981:
5980:
5974:
5970:
5958:
5944:
5943:
5937:
5933:
5921:
5914:
5908:
5897:
5896:
5895:
5891:
5889:
5886:
5885:
5881:
5854:
5853:
5841:
5827:
5826:
5814:
5804:
5791:
5790:
5789:
5780:
5769:
5768:
5767:
5765:
5762:
5761:
5738:
5724:
5720:
5716:
5710:
5709:
5708:
5694:
5693:
5682:
5669:
5668:
5667:
5653:
5652:
5650:
5647:
5646:
5626:
5625:
5614:
5602:
5601:
5589:
5577:
5573:
5562:
5559:
5558:
5541:
5540:
5529:
5517:
5516:
5504:
5496:
5493:
5492:
5486:
5462:
5451:
5450:
5449:
5443:
5438:
5437:
5428:
5417:
5416:
5415:
5413:
5410:
5409:
5387:
5376:
5375:
5374:
5369:
5354:
5343:
5342:
5341:
5339:
5336:
5335:
5317:
5310:
5306:
5294:
5290:
5284:
5283:
5282:
5275:
5274:
5269:
5263:
5262:
5257:
5247:
5246:
5236:
5223:
5219:
5207:
5203:
5197:
5196:
5195:
5193:
5190:
5189:
5180:
5151:
5147:
5135:
5131:
5119:
5115:
5103:
5099:
5078:
5071:
5067:
5066:
5065:
5060:
5057:
5056:
5034:
5019:
5015:
5003:
4999:
4989:
4983:
4979:
4961:
4957:
4945:
4941:
4932:
4928:
4926:
4923:
4922:
4892:
4888:
4867:
4860:
4856:
4855:
4854:
4849:
4839:
4833:
4829:
4828:
4826:
4817:
4813:
4811:
4808:
4807:
4784:
4780:
4778:
4775:
4774:
4753:
4749:
4742:
4740:
4737:
4736:
4718:
4713:
4701:
4697:
4691:
4687:
4669:
4665:
4653:
4649:
4647:
4644:
4643:
4616:
4612:
4607:
4604:
4603:
4580:
4576:
4571:
4568:
4567:
4559:
4556:
4526:
4522:
4507:
4503:
4494:
4490:
4488:
4485:
4484:
4477:
4466:
4438:
4434:
4422:
4418:
4409:
4405:
4403:
4400:
4399:
4370:
4366:
4364:
4361:
4360:
4341:
4327:
4324:
4323:
4303:
4299:
4281:
4277:
4268:
4264:
4243:
4239:
4230:
4226:
4205:
4201:
4177:
4173:
4155:
4151:
4142:
4138:
4132:
4128:
4107:
4103:
4091:
4087:
4069:
4065:
4063:
4060:
4059:
4052:
4018:
4014:
4002:
3998:
3989:
3985:
3983:
3980:
3979:
3955:
3952:
3951:
3934:
3930:
3928:
3925:
3924:
3890:
3886:
3884:
3881:
3880:
3877:
3844:
3841:
3840:
3823:
3819:
3808:
3804:
3792:
3788:
3786:
3774:
3768:
3765:
3764:
3758:Binet's formula
3738:
3732:
3728:
3716:
3712:
3710:
3707:
3706:
3689:
3685:
3683:
3680:
3679:
3662:
3658:
3656:
3653:
3652:
3622:
3618:
3606:
3602:
3600:
3588:
3582:
3579:
3578:
3560:
3557:
3556:
3549:Johannes Kepler
3546:
3515:
3501:
3499:
3484:
3480:
3475:
3472:
3471:
3468:
3460:
3456:
3446:
3439:
3406:
3402:
3397:
3394:
3393:
3390:
3382:
3364:
3359:
3353:
3349:
3347:
3344:
3343:
3336:
3329:
3286:
3282:
3280:
3277:
3276:
3239:
3236:
3235:
3206:
3202:
3197:
3179:
3175:
3152:
3116:
3112:
3110:
3107:
3106:
3061:
3042:
3033:
3029:
3028:
3024:
2987:
2986:
2982:
2980:
2977:
2976:
2972:
2925:
2916:
2912:
2911:
2907:
2890:
2887:
2886:
2882:
2875:
2868:
2864:
2819:
2815:
2813:
2809:
2800:
2796:
2794:
2791:
2790:
2762:
2758:
2756:
2754:
2751:
2750:
2745:is the closest
2743:
2735:
2728:
2707:
2687:
2683:
2681:
2677:
2675:
2672:
2671:
2668:
2650:
2649:
2632:
2628:
2616:
2612:
2611:
2609:
2602:
2596:
2595:
2575:
2571:
2562:
2558:
2557:
2555:
2548:
2541:
2539:
2536:
2535:
2514:
2510:
2498:
2494:
2482:
2478:
2476:
2473:
2472:
2467:
2461:
2458:
2452:
2405:
2389:
2384:
2376:
2373:
2372:
2351:
2350:
2340:
2322:
2321:
2311:
2298:
2294:
2292:
2289:
2288:
2282:
2278:
2275:
2267:
2263:
2257:
2253:
2247:
2243:
2239:
2223:
2222:
2207:
2203:
2188:
2184:
2175:
2174:
2162:
2158:
2140:
2136:
2118:
2114:
2096:
2092:
2080:
2079:
2064:
2060:
2045:
2041:
2017:
2013:
1998:
1994:
1979:
1978:
1972:
1968:
1956:
1952:
1942:
1936:
1932:
1928:
1926:
1923:
1922:
1901:
1897:
1885:
1881:
1869:
1865:
1863:
1860:
1859:
1853:
1849:
1833:
1832:
1817:
1813:
1798:
1794:
1787:
1781:
1777:
1774:
1773:
1758:
1754:
1739:
1735:
1728:
1722:
1718:
1714:
1712:
1709:
1708:
1702:
1698:
1672:
1668:
1653:
1649:
1640:
1636:
1634:
1631:
1630:
1601:
1597:
1595:
1592:
1591:
1587:
1583:
1551:
1541:
1537:
1519:
1515:
1514:
1512:
1492:
1488:
1470:
1466:
1465:
1463:
1454:
1450:
1448:
1445:
1444:
1422:
1418:
1407:
1404:
1403:
1361:
1330:
1323:
1321:
1313:
1310:
1309:
1299:
1256:
1249:
1247:
1239:
1236:
1235:
1203:
1199:
1190:
1186:
1185:
1183:
1167:
1160:
1156:
1147:
1143:
1142:
1140:
1131:
1127:
1125:
1122:
1121:
1104:Binet's formula
1088:
1083:
1077:
1069:
1039:
1035:
1028:
1021:
1017:
965:
942:
905:
896:
893:
885:
882:
873:
869:
851:
842:
838:
823:
817:
814:
808:
805:
799:
792:
786:
781:
707:
701:
697:
691:
687:
681:
677:
671:
667:
661:
657:
651:
647:
641:
637:
631:
627:
621:
617:
611:
607:
601:
597:
591:
587:
581:
577:
571:
567:
561:
557:
551:
547:
541:
537:
531:
527:
521:
517:
511:
500:
495:
485:
462:
458:
443:
439:
430:
426:
424:
421:
420:
394:
390:
381:
377:
375:
372:
371:
348:
344:
342:
339:
338:
328:
305:
301:
286:
282:
273:
269:
267:
264:
263:
237:
233:
217:
213:
211:
208:
207:
185:
177:Lucas sequences
164:
160:
156:
153:Binet's formula
122:Fibonacci cubes
48:
45:
40:
39:
24:
17:
12:
11:
5:
28920:
28910:
28909:
28904:
28887:
28886:
28884:
28883:
28882:
28881:
28869:
28864:
28859:
28853:
28851:
28847:
28846:
28844:
28843:
28838:
28832:
28830:
28826:
28825:
28823:
28822:
28814:
28805:
28803:
28799:
28798:
28791:
28790:
28783:
28776:
28768:
28759:
28758:
28756:
28755:
28744:
28741:
28740:
28738:
28737:
28732:
28727:
28722:
28717:
28712:
28706:
28704:
28698:
28697:
28695:
28694:
28689:
28687:Fourier series
28684:
28679:
28674:
28672:Puiseux series
28669:
28667:Laurent series
28664:
28659:
28654:
28648:
28646:
28642:
28641:
28638:
28637:
28635:
28634:
28629:
28624:
28619:
28614:
28609:
28604:
28599:
28594:
28589:
28583:
28581:
28577:
28576:
28574:
28573:
28568:
28563:
28558:
28552:
28550:
28543:
28539:
28538:
28535:
28534:
28532:
28531:
28526:
28521:
28515:
28513:
28509:
28508:
28506:
28505:
28500:
28495:
28490:
28484:
28482:
28475:
28471:
28470:
28468:
28467:
28462:
28457:
28451:
28449:
28445:
28444:
28437:
28435:
28432:
28431:
28429:
28428:
28427:
28426:
28416:
28411:
28406:
28401:
28396:
28391:
28386:
28381:
28376:
28370:
28368:
28357:
28356:
28354:
28353:
28348:
28343:
28338:
28333:
28328:
28323:
28318:
28313:
28307:
28305:
28298:
28292:
28291:
28280:
28279:
28272:
28265:
28257:
28248:
28247:
28240:
28238:
28236:
28235:
28232:
28229:
28226:
28223:
28222:
28221:
28216:
28205:
28204:
28203:
28202:
28197:
28192:
28187:
28185:Section search
28182:
28177:
28172:
28167:
28162:
28157:
28147:
28141:
28138:
28137:
28134:Metallic means
28130:
28129:
28122:
28115:
28107:
28098:
28097:
28095:
28094:
28083:
28080:
28079:
28076:
28075:
28073:
28072:
28066:
28063:
28062:
28049:
28048:
28045:
28044:
28042:
28041:
28036:
28030:
28027:
28026:
28013:
28012:
28009:
28008:
28006:
28005:
28003:Sorting number
28000:
27998:Pancake number
27994:
27991:
27990:
27977:
27976:
27973:
27972:
27970:
27969:
27964:
27958:
27955:
27954:
27941:
27940:
27937:
27936:
27934:
27933:
27928:
27923:
27917:
27914:
27913:
27910:Binary numbers
27901:
27900:
27897:
27896:
27893:
27892:
27890:
27889:
27883:
27881:
27877:
27876:
27874:
27873:
27868:
27863:
27858:
27853:
27848:
27843:
27837:
27835:
27831:
27830:
27828:
27827:
27822:
27817:
27812:
27807:
27801:
27799:
27791:
27790:
27788:
27787:
27782:
27777:
27772:
27767:
27762:
27757:
27751:
27749:
27742:
27741:
27739:
27738:
27737:
27736:
27725:
27723:
27720:P-adic numbers
27716:
27715:
27712:
27711:
27709:
27708:
27707:
27706:
27696:
27691:
27686:
27681:
27676:
27671:
27666:
27661:
27655:
27653:
27649:
27648:
27646:
27645:
27639:
27637:
27636:Coding-related
27633:
27632:
27630:
27629:
27624:
27618:
27616:
27612:
27611:
27609:
27608:
27603:
27598:
27593:
27587:
27585:
27576:
27575:
27574:
27573:
27571:Multiplicative
27568:
27557:
27555:
27540:
27539:
27535:Numeral system
27526:
27525:
27522:
27521:
27519:
27518:
27513:
27508:
27503:
27498:
27493:
27488:
27483:
27478:
27473:
27468:
27463:
27458:
27453:
27448:
27443:
27437:
27434:
27433:
27415:
27414:
27411:
27410:
27407:
27406:
27404:
27403:
27398:
27392:
27390:
27384:
27383:
27381:
27380:
27375:
27370:
27365:
27359:
27357:
27351:
27350:
27348:
27347:
27342:
27337:
27332:
27327:
27325:Highly totient
27322:
27316:
27314:
27308:
27307:
27305:
27304:
27299:
27293:
27291:
27285:
27284:
27282:
27281:
27276:
27271:
27266:
27261:
27256:
27251:
27246:
27241:
27236:
27231:
27226:
27221:
27216:
27211:
27206:
27201:
27196:
27191:
27186:
27181:
27179:Almost perfect
27176:
27170:
27168:
27158:
27157:
27141:
27140:
27137:
27136:
27134:
27133:
27128:
27123:
27118:
27113:
27108:
27103:
27098:
27093:
27088:
27083:
27078:
27072:
27069:
27068:
27056:
27055:
27052:
27051:
27049:
27048:
27043:
27038:
27033:
27027:
27024:
27023:
27011:
27010:
27007:
27006:
27004:
27003:
26998:
26993:
26988:
26986:Stirling first
26983:
26978:
26973:
26968:
26963:
26958:
26953:
26948:
26943:
26938:
26933:
26928:
26923:
26918:
26913:
26908:
26902:
26899:
26898:
26888:
26887:
26884:
26883:
26880:
26879:
26876:
26875:
26873:
26872:
26867:
26862:
26856:
26854:
26847:
26841:
26840:
26837:
26836:
26834:
26833:
26827:
26825:
26819:
26818:
26816:
26815:
26810:
26805:
26800:
26795:
26790:
26784:
26782:
26776:
26775:
26773:
26772:
26767:
26762:
26757:
26752:
26746:
26744:
26735:
26729:
26728:
26725:
26724:
26722:
26721:
26716:
26711:
26706:
26701:
26696:
26691:
26686:
26681:
26676:
26670:
26668:
26662:
26661:
26659:
26658:
26653:
26648:
26643:
26638:
26633:
26628:
26623:
26618:
26612:
26610:
26601:
26591:
26590:
26578:
26577:
26574:
26573:
26571:
26570:
26565:
26560:
26555:
26550:
26545:
26539:
26536:
26535:
26525:
26524:
26521:
26520:
26518:
26517:
26512:
26507:
26502:
26497:
26491:
26488:
26487:
26477:
26476:
26473:
26472:
26470:
26469:
26464:
26459:
26454:
26449:
26444:
26439:
26434:
26428:
26425:
26424:
26411:
26410:
26407:
26406:
26404:
26403:
26398:
26393:
26388:
26383:
26377:
26374:
26373:
26363:
26362:
26359:
26358:
26356:
26355:
26350:
26345:
26340:
26335:
26330:
26325:
26319:
26316:
26315:
26301:
26300:
26297:
26296:
26294:
26293:
26288:
26283:
26278:
26273:
26268:
26263:
26258:
26253:
26248:
26243:
26238:
26233:
26228:
26222:
26219:
26218:
26205:
26204:
26196:
26195:
26188:
26181:
26173:
26167:
26166:
26158:
26140:
26125:
26120:
26114:
26100:
26099:External links
26097:
26096:
26095:
26090:
26077:
26065:
26060:
26052:Broadway Books
26038:
26032:
26019:
26014:
25992:
25986:
25970:
25964:
25951:
25945:
25926:
25923:
25920:
25919:
25917:, p. 193.
25907:
25905:, p. 176.
25895:
25851:
25811:
25795:
25783:
25745:
25736:
25711:
25705:
25670:
25668:, p. 112.
25658:
25632:
25626:
25604:
25587:
25567:
25555:
25553:, p. 110.
25543:
25519:
25501:
25484:
25478:
25460:
25419:
25396:
25380:
25366:
25351:
25336:(in Russian),
25322:
25316:
25295:
25276:
25256:
25230:
25203:
25197:
25179:
25167:
25129:
25104:
25070:
25034:
24999:
24987:
24975:
24963:
24957:
24936:
24911:
24894:
24856:
24826:
24776:
24750:
24698:
24679:
24645:
24639:
24621:
24593:
24561:
24536:
24524:
24472:
24468:
24462:
24459:
24456:
24452:
24434:
24407:
24389:
24383:
24361:
24351:(19): 539â41,
24333:
24304:
24288:
24246:
24216:
24211:978-0521898065
24210:
24190:
24184:
24166:
24154:
24132:
24108:
24073:
24059:
24053:
24035:
24010:
23985:
23973:
23971:, p. 156.
23961:
23955:
23929:
23919:
23899:
23888:
23867:
23846:
23819:
23813:
23795:
23768:
23756:
23725:
23713:
23698:
23684:Academic Press
23664:
23642:
23640:, p. 197.
23627:
23616:
23593:
23591:, p. 180.
23581:
23569:
23557:
23545:
23528:
23502:
23471:
23465:
23440:
23411:
23410:
23408:
23405:
23402:
23401:
23387:
23386:
23384:
23381:
23379:
23376:
23375:
23374:
23368:
23362:
23359:Fibonacci word
23356:
23351:
23343:
23340:
23339:
23338:
23324:
23318:
23311:
23303:
23280:
23270:
23235:
23232:
23213:
23210:
23205:
23201:
23176:
23173:
23168:
23164:
23139:
23136:
23131:
23127:
23102:
23099:
23094:
23090:
23067:
23064:
23059:
23055:
23018:
23006:
22995:
22987:
22986:
22983:
22982:in honeybees).
22882:
22877:
22874:
22871:
22865:
22862:
22855:
22851:
22846:
22843:
22837:
22834:
22695:
22692:
22691:
22690:
22685:
22680:
22673:planning poker
22669:
22643:
22636:
22633:Fibonacci heap
22629:
22622:
22607:
22585:
22554:
22551:
22550:
22549:
22546:conformal maps
22534:
22519:Fibonacci cube
22515:
22502:
22497:
22490:
22487:
22484:
22481:
22477:
22471:
22466:
22462:
22456:
22453:
22450:
22446:
22440:
22437:
22434:
22430:
22426:
22423:
22420:
22415:
22411:
22405:
22402:
22399:
22395:
22389:
22385:
22381:
22366:right triangle
22358:
22343:
22336:
22319:
22305:
22293:
22273:
22261:
22235:
22223:
22208:
22154:
22136:
22124:
22086:
22085:
22082:
22081:
22076:
22071:
22066:
22063:
22062:
22057:
22052:
22047:
22042:
22037:
22021:
22014:
21998:
21995:
21992:
21981:domino tilings
21972:
21914:
21909:
21906:
21901:
21895:
21889:
21884:
21881:
21876:
21870:
21864:
21859:
21856:
21851:
21836:
21835:
21832:
21831:
21826:
21821:
21816:
21813:
21812:
21807:
21802:
21797:
21792:
21789:
21788:
21783:
21756:
21752:
21729:
21725:
21719:
21715:
21709:
21704:
21701:
21698:
21694:
21690:
21687:
21684:
21679:
21675:
21671:
21668:
21665:
21662:
21657:
21654:
21651:
21647:
21643:
21640:
21637:
21632:
21628:
21624:
21621:
21618:
21615:
21610:
21606:
21602:
21599:
21596:
21593:
21590:
21587:
21582:
21578:
21574:
21571:
21568:
21560:
21556:
21552:
21549:
21546:
21543:
21539:
21517:
21511:
21506:
21502:
21499:
21496:
21493:
21490:
21484:
21475:
21470:
21466:
21463:
21460:
21454:
21448:
21445:
21442:
21438:
21434:
21429:
21425:
21392:
21389:
21387:
21384:
21383:
21382:
21375:
21345:Perrin numbers
21337:
21321:
21311:
21303:
21293:
21281:
21271:
21263:
21254:
21244:
21233:
21226:
21196:Main article:
21193:
21190:
21162:Pisano periods
21135:Main article:
21132:
21126:
21093:
21090:
21087:
21084:
21081:
21078:
21073:
21068:
21063:
21059:
21052:
21049:
21046:
21041:
21037:
21030:
21027:
21024:
21019:
21015:
21008:
21005:
21002:
20999:
20996:
20991:
20986:
20981:
20977:
20970:
20967:
20964:
20959:
20955:
20948:
20945:
20942:
20937:
20933:
20926:
20923:
20920:
20915:
20911:
20904:
20901:
20898:
20893:
20889:
20870:
20859:
20836:
20833:
20829:
20826:
20821:
20818:
20815:
20812:
20807:
20800:
20796:
20790:
20777:
20774:
20770:
20767:
20762:
20759:
20756:
20753:
20748:
20741:
20737:
20731:
20712:
20709:
20704:
20700:
20691:
20688:
20683:
20679:
20659:
20656:
20652:
20648:
20645:
20640:
20632:
20629:
20621:
20616:
20612:
20605:
20602:
20595:
20592:
20589:
20585:
20581:
20578:
20573:
20565:
20562:
20554:
20549:
20545:
20538:
20535:
20528:
20525:
20522:
20519:
20514:
20506:
20503:
20495:
20454:
20451:
20447:
20444:
20439:
20436:
20433:
20430:
20425:
20418:
20414:
20408:
20395:
20392:
20388:
20385:
20380:
20377:
20374:
20371:
20368:
20363:
20356:
20352:
20346:
20327:
20324:
20319:
20315:
20306:
20303:
20298:
20294:
20274:
20271:
20267:
20263:
20260:
20255:
20247:
20244:
20236:
20231:
20227:
20220:
20217:
20210:
20207:
20204:
20201:
20197:
20193:
20190:
20185:
20177:
20174:
20166:
20161:
20157:
20150:
20147:
20140:
20137:
20134:
20131:
20126:
20118:
20115:
20107:
20066:
20063:
20059:
20056:
20051:
20048:
20045:
20042:
20037:
20030:
20026:
20020:
20007:
20004:
20000:
19997:
19992:
19989:
19986:
19983:
19978:
19971:
19967:
19961:
19942:
19939:
19934:
19930:
19921:
19918:
19913:
19909:
19889:
19886:
19882:
19878:
19875:
19870:
19862:
19859:
19851:
19846:
19842:
19835:
19832:
19825:
19822:
19819:
19815:
19811:
19808:
19803:
19795:
19792:
19784:
19779:
19775:
19768:
19765:
19758:
19755:
19752:
19749:
19744:
19736:
19733:
19725:
19684:
19681:
19677:
19674:
19669:
19666:
19663:
19660:
19657:
19652:
19645:
19641:
19635:
19622:
19619:
19615:
19612:
19607:
19604:
19601:
19598:
19595:
19590:
19583:
19579:
19573:
19554:
19551:
19546:
19542:
19533:
19530:
19525:
19521:
19501:
19498:
19495:
19491:
19487:
19484:
19479:
19471:
19468:
19460:
19455:
19451:
19444:
19441:
19434:
19431:
19428:
19425:
19421:
19417:
19414:
19409:
19401:
19398:
19390:
19385:
19381:
19374:
19371:
19364:
19361:
19358:
19355:
19350:
19342:
19339:
19331:
19289:
19284:
19280:
19277:
19273:
19270:
19265:
19262:
19259:
19251:
19248:
19245:
19241:
19238:
19232:
19228:
19225:
19220:
19212:
19209:
19201:
19196:
19192:
19185:
19182:
19176:
19175:
19171:
19168:
19164:
19161:
19156:
19153:
19150:
19142:
19139:
19136:
19132:
19129:
19123:
19119:
19116:
19111:
19103:
19100:
19092:
19087:
19083:
19076:
19073:
19067:
19066:
19064:
19059:
19054:
19046:
19042:
19039:
19036:
19030:
19024:
18988:
18984:
18979:
18975:
18970:
18967:
18962:
18959:
18953:
18948:
18945:
18940:
18936:
18933:
18929:
18896:
18893:
18890:
18888:
18884:
18880:
18876:
18874:
18871:
18868:
18865:
18863:
18859:
18855:
18851:
18849:
18846:
18843:
18840:
18837:
18835:
18831:
18823:
18820:
18812:
18807:
18806:
18803:
18800:
18797:
18794:
18792:
18788:
18784:
18780:
18778:
18775:
18772:
18769:
18767:
18763:
18759:
18755:
18753:
18750:
18747:
18744:
18741:
18739:
18735:
18727:
18724:
18716:
18711:
18710:
18707:
18704:
18701:
18698:
18696:
18692:
18688:
18684:
18682:
18679:
18676:
18673:
18671:
18667:
18659:
18656:
18648:
18643:
18642:
18639:
18636:
18633:
18630:
18628:
18624:
18620:
18616:
18614:
18611:
18608:
18605:
18603:
18599:
18595:
18591:
18589:
18586:
18583:
18580:
18577:
18575:
18571:
18563:
18560:
18552:
18547:
18546:
18543:
18540:
18537:
18534:
18532:
18528:
18524:
18520:
18518:
18515:
18512:
18509:
18507:
18503:
18499:
18495:
18493:
18490:
18487:
18484:
18481:
18479:
18475:
18467:
18464:
18456:
18451:
18450:
18428:
18424:
18421:
18417:
18414:
18409:
18406:
18400:
18395:
18392:
18387:
18383:
18380:
18376:
18364:
18361:
18357:
18354:
18348:
18343:
18340:
18335:
18331:
18326:
18322:
18293:
18288:
18284:
18281:
18277:
18274:
18269:
18266:
18263:
18260:
18252:
18250:
18247:
18244:
18243:
18239:
18236:
18232:
18229:
18224:
18221:
18218:
18215:
18207:
18205:
18202:
18201:
18198:
18195:
18192:
18184:
18182:
18179:
18178:
18176:
18171:
18167:
18162:
18159:
18154:
18131:
18123:
18120:
18112:
18087:
18078:
18065:
18056:
18047:
18040:
18032:
18031:Prime divisors
18029:
18021:perfect number
17985:
17964:
17953:
17942:
17933:
17932:
17913:Main article:
17910:
17907:
17880:
17876:
17873:
17869:
17866:
17859:
17853:
17847:
17844:
17842:
17839:
17838:
17835:
17832:
17830:
17827:
17826:
17824:
17818:
17813:
17805:
17802:
17799:
17795:
17791:
17787:
17783:
17779:
17778:
17773:
17769:
17765:
17761:
17758:
17755:
17751:
17747:
17746:
17744:
17720:
17668:
17662:
17657:
17654:
17649:
17644:
17640:
17636:
17632:
17629:
17618:primality test
17613:
17610:
17598:
17592:
17587:
17584:
17579:
17574:
17570:
17566:
17562:
17559:
17527:
17522:
17517:
17514:
17511:
17507:
17503:
17500:
17497:
17494:
17491:
17488:
17484:
17481:
17476:
17473:
17470:
17467:
17464:
17463:
17460:
17455:
17452:
17449:
17445:
17441:
17438:
17435:
17432:
17429:
17426:
17422:
17419:
17414:
17411:
17408:
17405:
17402:
17401:
17398:
17393:
17389:
17385:
17382:
17379:
17376:
17374:
17371:
17368:
17365:
17364:
17362:
17345:
17324:
17304:
17268:
17267:
17256:
17253:
17250:
17245:
17242:
17239:
17235:
17231:
17226:
17223:
17220:
17216:
17212:
17209:
17206:
17203:
17198:
17195:
17192:
17188:
17184:
17179:
17175:
17171:
17168:
17165:
17162:
17157:
17154:
17151:
17147:
17143:
17138:
17134:
17130:
17127:
17104:
17101:
17096:
17092:
17071:
17068:
17063:
17059:
17020:
17017:
17014:
17011:
17008:
17005:
17002:
16999:
16996:
16993:
16989:
16985:
16982:
16979:
16976:
16971:
16967:
16963:
16958:
16954:
16950:
16945:
16941:
16937:
16934:
16917:
16899:
16896:
16891:
16887:
16874:
16871:
16869:
16866:
16854:
16845:
16841:
16836:
16830:
16827:
16822:
16818:
16813:
16807:
16804:
16801:
16792:
16788:
16783:
16779:
16772:
16767:
16764:
16761:
16757:
16732:
16727:
16721:
16716:
16713:
16707:
16698:
16694:
16689:
16685:
16678:
16673:
16670:
16667:
16663:
16626:
16623:
16622:3.359885666243
16620:
16613:
16610:
16606:
16602:
16595:
16590:
16587:
16584:
16580:
16576:
16569:
16566:
16563:
16560:
16556:
16552:
16545:
16540:
16537:
16534:
16530:
16526:
16519:
16515:
16511:
16504:
16499:
16496:
16493:
16489:
16461:
16456:
16447:
16444:
16440:
16436:
16433:
16427:
16424:
16420:
16414:
16406:
16403:
16399:
16395:
16392:
16386:
16383:
16379:
16372:
16366:
16361:
16354:
16351:
16347:
16343:
16319:
16311:
16307:
16303:
16300:
16294:
16290:
16282:
16277:
16274:
16271:
16267:
16263:
16260:
16257:
16254:
16251:
16240:Lambert series
16226:
16222:
16217:
16213:
16209:
16206:
16203:
16200:
16195:
16191:
16187:
16184:
16180:
16174:
16169:
16162:
16159:
16155:
16151:
16144:
16139:
16136:
16133:
16129:
16106:
16101:
16097:
16094:
16089:
16081:
16073:
16066:
16062:
16054:
16049:
16046:
16043:
16039:
16031:
16028:
16025:
16021:
16017:
16014:
16011:
16003:
15998:
15995:
15992:
15988:
15957:
15952:
15948:
15942:
15934:
15931:
15928:
15925:
15921:
15917:
15914:
15910:
15903:
15898:
15895:
15892:
15888:
15865:
15861:
15857:
15854:
15849:
15844:
15838:
15832:
15827:
15824:
15818:
15815:
15811:
15803:
15799:
15795:
15790:
15785:
15779:
15773:
15768:
15765:
15759:
15756:
15752:
15744:
15740:
15735:
15728:
15725:
15720:
15713:
15706:
15702:
15696:
15689:
15684:
15681:
15678:
15674:
15651:
15646:
15641:
15635:
15629:
15624:
15621:
15615:
15612:
15608:
15600:
15596:
15589:
15585:
15579:
15572:
15569:
15566:
15563:
15559:
15555:
15548:
15543:
15540:
15537:
15533:
15512:
15509:
15497:
15494:
15491:
15488:
15483:
15480:
15475:
15470:
15467:
15462:
15459:
15456:
15453:
15450:
15430:
15427:
15424:
15421:
15401:
15379:
15375:
15369:
15366:
15361:
15357:
15352:
15349:
15346:
15343:
15340:
15337:
15323:satisfies the
15312:
15309:
15306:
15303:
15278:
15274:
15270:
15266:
15263:
15259:
15255:
15252:
15249:
15246:
15218:
15212:
15207:
15204:
15200:
15193:
15190:
15184:
15181:
15160:
15154:
15149:
15146:
15142:
15135:
15132:
15126:
15123:
15102:
15095:
15092:
15089:
15086:
15082:
15077:
15071:
15068:
15065:
15062:
15058:
15052:
15045:
15041:
15036:
15033:
15030:
15027:
15024:
14997:
14994:
14991:
14969:
14965:
14938:
14935:
14932:
14929:
14927:
14925:
14920:
14916:
14912:
14907:
14904:
14901:
14897:
14893:
14888:
14885:
14882:
14878:
14874:
14869:
14865:
14861:
14856:
14851:
14848:
14845:
14841:
14837:
14832:
14828:
14824:
14821:
14816:
14812:
14808:
14805:
14800:
14796:
14792:
14789:
14786:
14784:
14782:
14777:
14773:
14767:
14764:
14761:
14757:
14751:
14746:
14743:
14740:
14736:
14732:
14727:
14723:
14717:
14714:
14711:
14707:
14701:
14696:
14693:
14690:
14686:
14682:
14677:
14673:
14667:
14663:
14657:
14652:
14649:
14646:
14642:
14638:
14635:
14633:
14631:
14626:
14623:
14620:
14616:
14610:
14606:
14600:
14595:
14592:
14589:
14585:
14581:
14576:
14573:
14570:
14566:
14560:
14556:
14550:
14545:
14542:
14539:
14535:
14531:
14526:
14522:
14516:
14512:
14506:
14501:
14498:
14495:
14491:
14487:
14484:
14482:
14480:
14477:
14474:
14471:
14468:
14463:
14459:
14455:
14452:
14449:
14446:
14443:
14440:
14439:
14419:
14414:
14410:
14406:
14403:
14400:
14397:
14394:
14372:
14364:
14360:
14356:
14353:
14350:
14347:
14343:
14338:
14335:
14332:
14329:
14326:
14304:
14301:
14297:
14293:
14290:
14286:
14282:
14278:
14257:
14247:complex number
14232:
14229:
14226:
14221:
14217:
14213:
14210:
14205:
14201:
14197:
14194:
14189:
14185:
14181:
14178:
14173:
14169:
14165:
14162:
14159:
14156:
14153:
14148:
14144:
14138:
14134:
14128:
14123:
14120:
14117:
14113:
14109:
14106:
14103:
14100:
14097:
14076:
14073:
14047:
14042:
14039:
14036:
14029:
14026:
14023:
14019:
14011:
14004:
14000:
13992:
13989:
13986:
13982:
13975:
13970:
13967:
13962:
13954:
13949:
13946:
13943:
13939:
13935:
13930:
13927:
13924:
13921:
13917:
13892:
13887:
13884:
13881:
13874:
13871:
13868:
13864:
13856:
13849:
13845:
13837:
13834:
13831:
13827:
13820:
13815:
13812:
13807:
13799:
13794:
13791:
13788:
13784:
13780:
13775:
13772:
13769:
13766:
13762:
13723:
13717:
13710:
13707:
13704:
13700:
13694:
13691:
13686:
13679:
13675:
13668:
13662:
13655:
13651:
13645:
13642:
13638:
13632:
13625:
13621:
13615:
13612:
13607:
13600:
13597:
13594:
13590:
13583:
13577:
13574:
13571:
13567:
13561:
13557:
13553:
13550:
13545:
13542:
13538:
13516:
13509:
13505:
13499:
13494:
13490:
13484:
13477:
13474:
13471:
13467:
13461:
13458:
13453:
13446:
13443:
13440:
13436:
13430:
13425:
13422:
13419:
13416:
13412:
13390:
13383:
13379:
13373:
13368:
13361:
13357:
13349:
13346:
13343:
13339:
13335:
13332:
13327:
13320:
13317:
13314:
13310:
13304:
13299:
13296:
13293:
13290:
13286:
13261:
13257:
13251:
13247:
13243:
13240:
13237:
13234:
13231:
13226:
13219:
13215:
13209:
13206:
13201:
13198:
13195:
13191:
13185:
13182:
13179:
13175:
13169:
13165:
13161:
13158:
13153:
13146:
13142:
13136:
13133:
13128:
13125:
13121:
13093:
13075:
13071:
13065:
13061:
13057:
13053:
13047:
13044:
13041:
13037:
13033:
13028:
13025:
13022:
13018:
13013:
13007:
13003:
12999:
12994:
12987:
12984:
12981:
12977:
12971:
12966:
12959:
12956:
12953:
12949:
12943:
12938:
12935:
12931:
12909:
12906:
12903:
12899:
12893:
12889:
12885:
12882:
12879:
12876:
12871:
12867:
12861:
12858:
12855:
12851:
12847:
12842:
12839:
12836:
12832:
12826:
12822:
12810:
12807:
12793:
12786:
12782:
12774:
12771:
12768:
12764:
12760:
12757:
12754:
12751:
12746:
12743:
12740:
12736:
12730:
12727:
12724:
12720:
12716:
12711:
12704:
12700:
12675:
12672:
12669:
12665:
12661:
12658:
12655:
12652:
12647:
12644:
12641:
12637:
12631:
12628:
12625:
12621:
12617:
12612:
12605:
12601:
12583:Main article:
12580:
12577:
12572:
12569:
12554:
12549:
12546:
12543:
12540:
12520:
12517:
12514:
12511:
12508:
12484:
12481:
12478:
12473:
12470:
12467:
12463:
12459:
12454:
12449:
12446:
12444:
12442:
12439:
12436:
12433:
12430:
12427:
12424:
12419:
12416:
12413:
12409:
12405:
12400:
12397:
12394:
12390:
12386:
12383:
12381:
12379:
12373:
12370:
12365:
12362:
12357:
12354:
12351:
12347:
12343:
12340:
12337:
12332:
12329:
12326:
12322:
12318:
12312:
12309:
12307:
12305:
12299:
12296:
12291:
12288:
12283:
12280:
12277:
12273:
12266:
12260:
12257:
12252:
12249:
12244:
12241:
12238:
12234:
12227:
12224:
12222:
12220:
12214:
12211:
12208:
12203:
12200:
12195:
12192:
12189:
12185:
12178:
12172:
12169:
12166:
12161:
12158:
12153:
12150:
12147:
12143:
12136:
12133:
12131:
12129:
12125:
12119:
12115:
12111:
12108:
12105:
12100:
12096:
12092:
12089:
12086:
12083:
12079:
12075:
12070:
12066:
12062:
12059:
12056:
12051:
12047:
12043:
12040:
12037:
12035:
12032:
12029:
12028:
12006:
11984:
11981:
11976:
11973:
11970:
11966:
11962:
11957:
11953:
11947:
11942:
11939:
11936:
11932:
11908:
11903:
11899:
11895:
11890:
11886:
11882:
11877:
11873:
11867:
11853:
11850:
11836:
11833:
11830:
11826:
11820:
11817:
11814:
11810:
11806:
11801:
11796:
11792:
11786:
11783:
11780:
11775:
11772:
11769:
11765:
11744:
11738:
11735:
11732:
11728:
11724:
11719:
11715:
11710:
11704:
11701:
11698:
11694:
11690:
11685:
11678:
11675:
11672:
11668:
11662:
11657:
11652:
11648:
11642:
11637:
11634:
11631:
11627:
11604:
11601:
11598:
11594:
11588:
11584:
11580:
11575:
11570:
11566:
11560:
11555:
11552:
11549:
11545:
11522:
11515:
11512:
11509:
11505:
11480:
11477:
11472:
11469:
11466:
11462:
11458:
11453:
11449:
11443:
11440:
11437:
11432:
11429:
11426:
11422:
11402:
11399:
11396:
11385:
11384:
11373:
11370:
11365:
11362:
11359:
11355:
11351:
11346:
11343:
11340:
11336:
11332:
11327:
11324:
11321:
11317:
11313:
11308:
11304:
11298:
11293:
11290:
11287:
11283:
11257:
11254:
11251:
11247:
11226:
11223:
11218:
11215:
11212:
11208:
11204:
11199:
11195:
11189:
11184:
11181:
11178:
11174:
11154:
11151:
11139:
11131:
11127:
11123:
11120:
11117:
11114:
11110:
11083:
11079:
11073:
11069:
11063:
11058:
11055:
11052:
11048:
11020:
11017:
11014:
10986:
10966:
10959:
10955:
10951:
10948:
10943:
10940:
10937:
10906:
10902:
10899:
10895:
10889:
10885:
10881:
10869:
10866:
10840:
10836:
10832:
10829:
10826:
10823:
10820:
10815:
10812:
10809:
10805:
10801:
10796:
10792:
10769:
10766:
10763:
10759:
10755:
10750:
10746:
10711:
10707:
10684:
10681:
10678:
10674:
10668:
10664:
10660:
10655:
10650:
10646:
10640:
10635:
10632:
10629:
10625:
10591:
10588:
10584:
10549:
10546:
10543:
10540:
10536:
10511:
10508:
10503:
10500:
10497:
10494:
10490:
10486:
10481:
10478:
10474:
10468:
10463:
10460:
10457:
10453:
10430:
10427:
10423:
10419:
10414:
10411:
10408:
10405:
10401:
10395:
10392:
10389:
10384:
10381:
10378:
10374:
10350:
10347:
10342:
10339:
10336:
10332:
10328:
10323:
10319:
10313:
10308:
10305:
10302:
10298:
10277:
10274:
10269:
10265:
10253:
10252:
10240:
10236:
10233:
10230:
10227:
10224:
10221:
10218:
10215:
10212:
10209:
10206:
10203:
10200:
10196:
10192:
10188:
10184:
10181:
10178:
10175:
10172:
10169:
10166:
10163:
10160:
10157:
10154:
10151:
10148:
10145:
10142:
10138:
10134:
10131:
10128:
10125:
10122:
10117:
10114:
10111:
10107:
10103:
10098:
10094:
10090:
10085:
10082:
10079:
10075:
10048:
10045:
10042:
10039:
10036:
10033:
10030:
10027:
10024:
10021:
10018:
10015:
10012:
10009:
10006:
10003:
10000:
9997:
9994:
9991:
9988:
9985:
9982:
9979:
9976:
9973:
9970:
9967:
9964:
9944:
9941:
9938:
9935:
9932:
9929:
9926:
9923:
9920:
9917:
9914:
9911:
9908:
9905:
9902:
9899:
9896:
9893:
9890:
9887:
9884:
9881:
9861:
9858:
9855:
9833:
9830:
9825:
9822:
9819:
9815:
9811:
9806:
9802:
9796:
9791:
9788:
9785:
9781:
9743:
9739:
9716:
9713:
9710:
9706:
9702:
9697:
9694:
9691:
9687:
9664:
9661:
9658:
9654:
9631:
9628:
9625:
9621:
9600:
9597:
9594:
9574:
9571:
9568:
9547:
9543:
9540:
9537:
9534:
9531:
9528:
9525:
9522:
9519:
9516:
9513:
9510:
9507:
9504:
9501:
9498:
9495:
9492:
9489:
9486:
9483:
9479:
9475:
9471:
9467:
9464:
9461:
9458:
9455:
9452:
9449:
9446:
9443:
9440:
9437:
9434:
9431:
9428:
9425:
9422:
9419:
9416:
9413:
9410:
9407:
9403:
9399:
9394:
9390:
9367:
9363:
9340:
9337:
9334:
9330:
9326:
9321:
9318:
9315:
9311:
9307:
9302:
9298:
9286:
9285:
9273:
9269:
9266:
9263:
9260:
9257:
9254:
9251:
9248:
9245:
9242:
9239:
9236:
9233:
9230:
9227:
9224:
9221:
9218:
9215:
9212:
9209:
9206:
9203:
9200:
9197:
9194:
9191:
9188:
9185:
9182:
9179:
9176:
9173:
9170:
9167:
9164:
9161:
9158:
9155:
9152:
9149:
9145:
9141:
9138:
9135:
9130:
9126:
9115:
9103:
9099:
9096:
9093:
9090:
9087:
9084:
9081:
9078:
9075:
9072:
9069:
9066:
9063:
9060:
9057:
9054:
9051:
9048:
9045:
9042:
9039:
9035:
9031:
9028:
9025:
9020:
9016:
9005:
8993:
8989:
8986:
8983:
8980:
8977:
8974:
8971:
8968:
8965:
8962:
8959:
8955:
8951:
8948:
8945:
8940:
8936:
8925:
8913:
8909:
8906:
8903:
8900:
8897:
8893:
8889:
8886:
8883:
8878:
8874:
8863:
8851:
8847:
8844:
8841:
8838:
8834:
8830:
8827:
8824:
8819:
8815:
8804:
8792:
8788:
8785:
8781:
8777:
8774:
8771:
8766:
8762:
8729:
8724:
8721:
8718:
8713:
8692:
8689:
8684:
8680:
8659:
8656:
8651:
8647:
8624:
8620:
8599:
8596:
8593:
8571:
8567:
8550:
8547:
8545:
8542:
8500:
8495:
8491:
8487:
8482:
8478:
8474:
8469:
8466:
8463:
8459:
8455:
8452:
8449:
8446:
8444:
8442:
8437:
8433:
8429:
8424:
8420:
8416:
8411:
8408:
8405:
8401:
8397:
8394:
8391:
8388:
8386:
8384:
8379:
8375:
8371:
8366:
8363:
8360:
8356:
8352:
8347:
8344:
8341:
8337:
8333:
8330:
8327:
8325:
8318:
8315:
8307:
8304:
8300:
8296:
8295:
8290:
8283:
8280:
8277:
8273:
8267:
8262:
8255:
8251:
8245:
8242:
8240:
8236:
8233:
8230:
8227:
8223:
8219:
8218:
8182:
8177:
8174:
8171:
8167:
8163:
8160:
8158:
8154:
8150:
8144:
8141:
8138:
8134:
8130:
8125:
8122:
8119:
8115:
8109:
8105:
8101:
8100:
8097:
8092:
8089:
8086:
8083:
8080:
8076:
8072:
8069:
8067:
8062:
8059:
8056:
8052:
8044:
8041:
8038:
8034:
8029:
8023:
8019:
8011:
8007:
8002:
8001:
7979:matrix product
7938:
7933:
7926:
7922:
7916:
7911:
7908:
7905:
7901:
7895:
7892:
7889:
7885:
7881:
7876:
7872:
7868:
7865:
7862:
7814:
7809:
7801:
7798:
7795:
7791:
7787:
7783:
7779:
7775:
7774:
7769:
7765:
7761:
7757:
7754:
7751:
7747:
7743:
7742:
7740:
7735:
7730:
7724:
7718:
7715:
7713:
7710:
7709:
7706:
7703:
7701:
7698:
7697:
7695:
7675:
7665:
7642:
7632:
7623:
7597:
7579:
7576:
7573:
7561:
7550:
7547:
7535:
7524:
7521:
7509:
7498:
7495:
7492:
7489:
7440:
7431:
7416:
7413:
7409:
7405:
7402:
7399:
7396:
7391:
7387:
7375:
7370:
7366:
7339:
7333:
7328:
7325:
7320:
7312:
7306:
7303:
7301:
7298:
7295:
7294:
7289:
7286:
7282:
7278:
7276:
7273:
7272:
7270:
7262:
7258:
7251:
7243:
7240:
7236:
7232:
7229:
7226:
7223:
7221:
7218:
7217:
7214:
7211:
7207:
7203:
7199:
7198:
7196:
7189:
7183:
7180:
7178:
7175:
7174:
7169:
7166:
7162:
7158:
7155:
7153:
7150:
7149:
7147:
7142:
7139:
7137:
7135:
7129:
7122:
7118:
7112:
7108:
7102:
7094:
7091:
7087:
7081:
7073:
7070:
7066:
7062:
7059:
7056:
7053:
7051:
7048:
7047:
7044:
7041:
7037:
7033:
7029:
7028:
7026:
7021:
7018:
7015:
7013:
7011:
7005:
6998:
6994:
6988:
6984:
6978:
6970:
6967:
6963:
6957:
6953:
6949:
6946:
6943:
6941:
6939:
6933:
6926:
6922:
6916:
6912:
6906:
6898:
6894:
6890:
6887:
6885:
6880:
6873:
6869:
6863:
6860:
6857:
6853:
6847:
6841:
6840:
6812:
6807:
6801:
6798:
6796:
6793:
6792:
6787:
6784:
6780:
6776:
6773:
6771:
6768:
6767:
6765:
6760:
6757:
6753:
6748:
6739:
6736:
6732:
6728:
6725:
6723:
6720:
6719:
6716:
6713:
6711:
6708:
6707:
6705:
6700:
6697:
6669:
6664:
6661:
6657:
6651:
6647:
6643:
6640:
6637:
6635:
6631:
6627:
6623:
6622:
6619:
6614:
6611:
6607:
6603:
6600:
6597:
6594:
6592:
6590:
6587:
6586:
6548:
6543:
6537:
6529:
6515:
6510:
6507:
6496:
6485:
6472:
6460:
6455:
6449:
6441:
6427:
6422:
6419:
6408:
6397:
6384:
6373:
6368:
6364:
6329:
6323:
6318:
6312:
6309:
6305:
6301:
6295:
6287:
6281:
6273:
6259:
6254:
6251:
6240:
6229:
6216:
6204:
6197:
6192:
6189:
6184:
6176:
6170:
6162:
6148:
6143:
6140:
6129:
6118:
6105:
6094:
6091:
6089:
6087:
6081:
6078:
6070:
6067:
6063:
6059:
6056:
6053:
6047:
6043:
6038:
6032:
6029:
6021:
6017:
6010:
6006:
6001:
5998:
5996:
5994:
5988:
5985:
5977:
5973:
5966:
5962:
5957:
5951:
5948:
5940:
5936:
5929:
5925:
5920:
5917:
5915:
5911:
5904:
5901:
5894:
5893:
5867:
5861:
5858:
5849:
5845:
5840:
5834:
5831:
5822:
5818:
5813:
5807:
5802:
5799:
5794:
5788:
5783:
5776:
5773:
5747:
5741:
5736:
5730:
5727:
5723:
5719:
5713:
5707:
5701:
5698:
5691:
5685:
5680:
5677:
5672:
5666:
5660:
5657:
5629:
5618:
5613:
5610:
5605:
5597:
5594:
5588:
5583:
5580:
5576:
5572:
5569:
5566:
5544:
5533:
5528:
5525:
5520:
5512:
5509:
5503:
5500:
5465:
5458:
5455:
5446:
5441:
5436:
5431:
5424:
5421:
5395:
5390:
5383:
5380:
5372:
5368:
5363:
5360:
5357:
5350:
5347:
5320:
5313:
5309:
5303:
5300:
5297:
5293:
5287:
5279:
5273:
5270:
5268:
5265:
5264:
5261:
5258:
5256:
5253:
5252:
5250:
5245:
5239:
5232:
5229:
5226:
5222:
5216:
5213:
5210:
5206:
5200:
5179:
5176:
5172:
5171:
5160:
5154:
5150:
5144:
5141:
5138:
5134:
5130:
5127:
5122:
5118:
5114:
5111:
5106:
5102:
5098:
5095:
5092:
5089:
5086:
5081:
5074:
5070:
5064:
5041:
5037:
5033:
5028:
5025:
5022:
5018:
5014:
5011:
5006:
5002:
4998:
4993:
4986:
4982:
4978:
4975:
4970:
4967:
4964:
4960:
4956:
4953:
4948:
4944:
4940:
4935:
4931:
4908:
4903:
4895:
4891:
4887:
4884:
4881:
4878:
4875:
4870:
4863:
4859:
4853:
4848:
4843:
4836:
4832:
4825:
4820:
4816:
4787:
4783:
4756:
4752:
4746:
4722:
4716:
4712:
4707:
4704:
4700:
4694:
4690:
4686:
4683:
4680:
4677:
4672:
4668:
4664:
4661:
4656:
4652:
4640:perfect square
4627:
4624:
4619:
4615:
4611:
4591:
4588:
4583:
4579:
4575:
4564:if and only if
4555:
4554:Identification
4552:
4540:
4535:
4532:
4529:
4525:
4521:
4516:
4513:
4510:
4506:
4502:
4497:
4493:
4475:
4452:
4447:
4444:
4441:
4437:
4433:
4430:
4425:
4421:
4417:
4412:
4408:
4387:
4384:
4381:
4378:
4373:
4369:
4348:
4344:
4340:
4337:
4334:
4331:
4311:
4306:
4302:
4298:
4295:
4290:
4287:
4284:
4280:
4276:
4271:
4267:
4263:
4260:
4257:
4252:
4249:
4246:
4242:
4238:
4233:
4229:
4225:
4222:
4219:
4214:
4211:
4208:
4204:
4200:
4197:
4194:
4191:
4188:
4185:
4180:
4176:
4172:
4169:
4164:
4161:
4158:
4154:
4150:
4145:
4141:
4135:
4131:
4127:
4124:
4121:
4116:
4113:
4110:
4106:
4102:
4099:
4094:
4090:
4086:
4083:
4078:
4075:
4072:
4068:
4032:
4027:
4024:
4021:
4017:
4013:
4010:
4005:
4001:
3997:
3992:
3988:
3959:
3937:
3933:
3910:
3907:
3904:
3901:
3898:
3893:
3889:
3876:
3873:
3872:
3871:
3848:
3826:
3822:
3818:
3811:
3807:
3801:
3798:
3795:
3791:
3783:
3780:
3777:
3773:
3745:
3741:
3735:
3731:
3727:
3724:
3719:
3715:
3692:
3688:
3665:
3661:
3638:
3635:
3632:
3625:
3621:
3615:
3612:
3609:
3605:
3597:
3594:
3591:
3587:
3567:
3564:
3545:
3542:
3530:
3524:
3521:
3518:
3513:
3510:
3507:
3504:
3498:
3495:
3492:
3487:
3483:
3479:
3464:
3427:
3423:
3420:
3417:
3414:
3409:
3405:
3401:
3386:
3368:
3362:
3356:
3352:
3334:
3328:
3325:
3312:
3309:
3306:
3303:
3300:
3297:
3294:
3289:
3285:
3264:
3261:
3258:
3255:
3252:
3249:
3246:
3243:
3223:
3220:
3217:
3214:
3209:
3205:
3200:
3196:
3193:
3190:
3187:
3182:
3178:
3174:
3171:
3168:
3165:
3162:
3159:
3155:
3151:
3148:
3145:
3142:
3139:
3136:
3133:
3130:
3127:
3124:
3119:
3115:
3094:
3091:
3088:
3085:
3079:
3075:
3071:
3068:
3064:
3060:
3057:
3054:
3051:
3046:
3041:
3036:
3032:
3027:
3023:
3020:
3017:
3014:
3008:
3005:
3002:
2999:
2996:
2993:
2990:
2985:
2969:floor function
2954:
2951:
2948:
2942:
2938:
2934:
2929:
2924:
2919:
2915:
2910:
2906:
2903:
2900:
2897:
2894:
2850:
2847:
2844:
2838:
2834:
2828:
2822:
2818:
2812:
2808:
2803:
2799:
2771:
2765:
2761:
2739:
2714:
2711:
2706:
2702:
2696:
2690:
2686:
2680:
2667:
2664:
2648:
2642:
2635:
2631:
2627:
2624:
2619:
2615:
2608:
2605:
2603:
2601:
2598:
2597:
2594:
2588:
2583:
2578:
2574:
2570:
2565:
2561:
2554:
2551:
2549:
2547:
2544:
2543:
2517:
2513:
2509:
2506:
2501:
2497:
2493:
2490:
2485:
2481:
2465:
2456:
2435:
2432:
2429:
2426:
2423:
2419:
2413:
2409:
2404:
2398:
2395:
2392:
2388:
2383:
2380:
2355:
2349:
2346:
2343:
2341:
2339:
2336:
2333:
2330:
2327:
2324:
2323:
2320:
2317:
2314:
2312:
2310:
2307:
2304:
2301:
2300:
2297:
2271:
2261:
2251:
2221:
2216:
2213:
2210:
2206:
2202:
2197:
2194:
2191:
2187:
2183:
2180:
2178:
2176:
2171:
2168:
2165:
2161:
2157:
2154:
2149:
2146:
2143:
2139:
2135:
2132:
2127:
2124:
2121:
2117:
2113:
2110:
2105:
2102:
2099:
2095:
2091:
2088:
2085:
2083:
2081:
2078:
2073:
2070:
2067:
2063:
2059:
2054:
2051:
2048:
2044:
2040:
2037:
2034:
2031:
2026:
2023:
2020:
2016:
2012:
2007:
2004:
2001:
1997:
1993:
1990:
1987:
1984:
1982:
1980:
1975:
1971:
1967:
1964:
1959:
1955:
1951:
1948:
1945:
1943:
1939:
1935:
1931:
1930:
1904:
1900:
1896:
1893:
1888:
1884:
1880:
1877:
1872:
1868:
1831:
1826:
1823:
1820:
1816:
1812:
1807:
1804:
1801:
1797:
1793:
1790:
1788:
1784:
1780:
1776:
1775:
1772:
1767:
1764:
1761:
1757:
1753:
1748:
1745:
1742:
1738:
1734:
1731:
1729:
1725:
1721:
1717:
1716:
1686:
1681:
1678:
1675:
1671:
1667:
1662:
1659:
1656:
1652:
1648:
1643:
1639:
1618:
1615:
1612:
1609:
1604:
1600:
1569:
1563:
1560:
1557:
1554:
1547:
1544:
1540:
1536:
1533:
1530:
1527:
1522:
1518:
1511:
1505:
1498:
1495:
1491:
1487:
1484:
1481:
1478:
1473:
1469:
1462:
1457:
1453:
1428:
1425:
1421:
1417:
1414:
1411:
1389:
1386:
1383:
1379:
1376:
1373:
1368:
1365:
1360:
1357:
1354:
1351:
1348:
1345:
1340:
1334:
1329:
1326:
1320:
1317:
1281:
1278:
1274:
1271:
1266:
1260:
1255:
1252:
1246:
1243:
1219:
1213:
1206:
1202:
1198:
1193:
1189:
1182:
1176:
1173:
1170:
1163:
1159:
1155:
1150:
1146:
1139:
1134:
1130:
1087:
1084:
1079:Main article:
1076:
1073:
1067:
1014:
1013:
1010:
1007:
1004:
941:
938:
900:
889:
877:
846:
821:
812:
803:
785:
782:
780:
777:
776:
775:
772:
771:
768:
765:
762:
759:
756:
753:
750:
747:
744:
741:
738:
735:
732:
729:
726:
723:
720:
717:
714:
710:
709:
705:
699:
695:
689:
685:
679:
675:
669:
665:
659:
655:
649:
645:
639:
635:
629:
625:
619:
615:
609:
605:
599:
595:
589:
585:
579:
575:
569:
565:
559:
555:
549:
545:
539:
535:
529:
525:
519:
515:
498:
471:
468:
465:
461:
457:
452:
449:
446:
442:
438:
433:
429:
408:
405:
402:
397:
393:
389:
384:
380:
359:
356:
351:
347:
314:
311:
308:
304:
300:
295:
292:
289:
285:
281:
276:
272:
251:
248:
245:
240:
236:
231:
228:
225:
220:
216:
184:
181:
155:expresses the
114:data structure
111:Fibonacci heap
64:A tiling with
58:
57:
43:
15:
9:
6:
4:
3:
2:
28919:
28908:
28905:
28903:
28900:
28899:
28897:
28880:
28879:
28875:
28874:
28873:
28870:
28868:
28865:
28863:
28860:
28858:
28855:
28854:
28852:
28848:
28842:
28839:
28837:
28834:
28833:
28831:
28827:
28820:
28819:
28815:
28812:
28811:
28807:
28806:
28804:
28800:
28796:
28789:
28784:
28782:
28777:
28775:
28770:
28769:
28766:
28754:
28746:
28745:
28742:
28736:
28733:
28731:
28728:
28726:
28723:
28721:
28718:
28716:
28713:
28711:
28708:
28707:
28705:
28703:
28699:
28693:
28690:
28688:
28685:
28683:
28680:
28678:
28675:
28673:
28670:
28668:
28665:
28663:
28660:
28658:
28655:
28653:
28652:Taylor series
28650:
28649:
28647:
28643:
28633:
28630:
28628:
28625:
28623:
28620:
28618:
28615:
28613:
28610:
28608:
28605:
28603:
28600:
28598:
28595:
28593:
28590:
28588:
28585:
28584:
28582:
28578:
28572:
28569:
28567:
28564:
28562:
28559:
28557:
28554:
28553:
28551:
28547:
28544:
28540:
28530:
28527:
28525:
28522:
28520:
28517:
28516:
28514:
28510:
28504:
28501:
28499:
28496:
28494:
28491:
28489:
28486:
28485:
28483:
28479:
28476:
28472:
28466:
28463:
28461:
28458:
28456:
28453:
28452:
28450:
28446:
28441:
28425:
28422:
28421:
28420:
28417:
28415:
28412:
28410:
28407:
28405:
28402:
28400:
28397:
28395:
28392:
28390:
28387:
28385:
28382:
28380:
28377:
28375:
28372:
28371:
28369:
28365:
28358:
28352:
28349:
28347:
28344:
28342:
28341:Powers of two
28339:
28337:
28334:
28332:
28329:
28327:
28326:Square number
28324:
28322:
28319:
28317:
28314:
28312:
28309:
28308:
28306:
28302:
28299:
28297:
28293:
28289:
28285:
28278:
28273:
28271:
28266:
28264:
28259:
28258:
28255:
28244:
28233:
28230:
28227:
28224:
28220:
28217:
28215:
28212:
28211:
28210:
28207:
28206:
28201:
28198:
28196:
28193:
28191:
28188:
28186:
28183:
28181:
28178:
28176:
28173:
28171:
28168:
28166:
28163:
28161:
28158:
28156:
28153:
28152:
28151:
28148:
28146:
28143:
28142:
28139:
28135:
28128:
28123:
28121:
28116:
28114:
28109:
28108:
28105:
28093:
28089:
28085:
28084:
28081:
28071:
28068:
28067:
28064:
28059:
28054:
28050:
28040:
28037:
28035:
28032:
28031:
28028:
28023:
28018:
28014:
28004:
28001:
27999:
27996:
27995:
27992:
27987:
27982:
27978:
27968:
27965:
27963:
27960:
27959:
27956:
27952:
27946:
27942:
27932:
27929:
27927:
27924:
27922:
27919:
27918:
27915:
27911:
27906:
27902:
27888:
27885:
27884:
27882:
27878:
27872:
27869:
27867:
27864:
27862:
27861:Polydivisible
27859:
27857:
27854:
27852:
27849:
27847:
27844:
27842:
27839:
27838:
27836:
27832:
27826:
27823:
27821:
27818:
27816:
27813:
27811:
27808:
27806:
27803:
27802:
27800:
27797:
27792:
27786:
27783:
27781:
27778:
27776:
27773:
27771:
27768:
27766:
27763:
27761:
27758:
27756:
27753:
27752:
27750:
27747:
27743:
27735:
27732:
27731:
27730:
27727:
27726:
27724:
27721:
27717:
27705:
27702:
27701:
27700:
27697:
27695:
27692:
27690:
27687:
27685:
27682:
27680:
27677:
27675:
27672:
27670:
27667:
27665:
27662:
27660:
27657:
27656:
27654:
27650:
27644:
27641:
27640:
27638:
27634:
27628:
27625:
27623:
27620:
27619:
27617:
27615:Digit product
27613:
27607:
27604:
27602:
27599:
27597:
27594:
27592:
27589:
27588:
27586:
27584:
27580:
27572:
27569:
27567:
27564:
27563:
27562:
27559:
27558:
27556:
27554:
27549:
27545:
27541:
27536:
27531:
27527:
27517:
27514:
27512:
27509:
27507:
27504:
27502:
27499:
27497:
27494:
27492:
27489:
27487:
27484:
27482:
27479:
27477:
27474:
27472:
27469:
27467:
27464:
27462:
27459:
27457:
27454:
27452:
27451:ErdĆsâNicolas
27449:
27447:
27444:
27442:
27439:
27438:
27435:
27430:
27426:
27420:
27416:
27402:
27399:
27397:
27394:
27393:
27391:
27389:
27385:
27379:
27376:
27374:
27371:
27369:
27366:
27364:
27361:
27360:
27358:
27356:
27352:
27346:
27343:
27341:
27338:
27336:
27333:
27331:
27328:
27326:
27323:
27321:
27318:
27317:
27315:
27313:
27309:
27303:
27300:
27298:
27295:
27294:
27292:
27290:
27286:
27280:
27277:
27275:
27272:
27270:
27269:Superabundant
27267:
27265:
27262:
27260:
27257:
27255:
27252:
27250:
27247:
27245:
27242:
27240:
27237:
27235:
27232:
27230:
27227:
27225:
27222:
27220:
27217:
27215:
27212:
27210:
27207:
27205:
27202:
27200:
27197:
27195:
27192:
27190:
27187:
27185:
27182:
27180:
27177:
27175:
27172:
27171:
27169:
27167:
27163:
27159:
27155:
27151:
27146:
27142:
27132:
27129:
27127:
27124:
27122:
27119:
27117:
27114:
27112:
27109:
27107:
27104:
27102:
27099:
27097:
27094:
27092:
27089:
27087:
27084:
27082:
27079:
27077:
27074:
27073:
27070:
27066:
27061:
27057:
27047:
27044:
27042:
27039:
27037:
27034:
27032:
27029:
27028:
27025:
27021:
27016:
27012:
27002:
26999:
26997:
26994:
26992:
26989:
26987:
26984:
26982:
26979:
26977:
26974:
26972:
26969:
26967:
26964:
26962:
26959:
26957:
26954:
26952:
26949:
26947:
26944:
26942:
26939:
26937:
26934:
26932:
26929:
26927:
26924:
26922:
26919:
26917:
26914:
26912:
26909:
26907:
26904:
26903:
26900:
26893:
26889:
26871:
26868:
26866:
26863:
26861:
26858:
26857:
26855:
26851:
26848:
26846:
26845:4-dimensional
26842:
26832:
26829:
26828:
26826:
26824:
26820:
26814:
26811:
26809:
26806:
26804:
26801:
26799:
26796:
26794:
26791:
26789:
26786:
26785:
26783:
26781:
26777:
26771:
26768:
26766:
26763:
26761:
26758:
26756:
26755:Centered cube
26753:
26751:
26748:
26747:
26745:
26743:
26739:
26736:
26734:
26733:3-dimensional
26730:
26720:
26717:
26715:
26712:
26710:
26707:
26705:
26702:
26700:
26697:
26695:
26692:
26690:
26687:
26685:
26682:
26680:
26677:
26675:
26672:
26671:
26669:
26667:
26663:
26657:
26654:
26652:
26649:
26647:
26644:
26642:
26639:
26637:
26634:
26632:
26629:
26627:
26624:
26622:
26619:
26617:
26614:
26613:
26611:
26609:
26605:
26602:
26600:
26599:2-dimensional
26596:
26592:
26588:
26583:
26579:
26569:
26566:
26564:
26561:
26559:
26556:
26554:
26551:
26549:
26546:
26544:
26543:Nonhypotenuse
26541:
26540:
26537:
26530:
26526:
26516:
26513:
26511:
26508:
26506:
26503:
26501:
26498:
26496:
26493:
26492:
26489:
26482:
26478:
26468:
26465:
26463:
26460:
26458:
26455:
26453:
26450:
26448:
26445:
26443:
26440:
26438:
26435:
26433:
26430:
26429:
26426:
26421:
26416:
26412:
26402:
26399:
26397:
26394:
26392:
26389:
26387:
26384:
26382:
26379:
26378:
26375:
26368:
26364:
26354:
26351:
26349:
26346:
26344:
26341:
26339:
26336:
26334:
26331:
26329:
26326:
26324:
26321:
26320:
26317:
26312:
26306:
26302:
26292:
26289:
26287:
26284:
26282:
26281:Perfect power
26279:
26277:
26274:
26272:
26271:Seventh power
26269:
26267:
26264:
26262:
26259:
26257:
26254:
26252:
26249:
26247:
26244:
26242:
26239:
26237:
26234:
26232:
26229:
26227:
26224:
26223:
26220:
26215:
26210:
26206:
26202:
26194:
26189:
26187:
26182:
26180:
26175:
26174:
26171:
26165:
26159:
26155:
26151:
26150:
26145:
26141:
26139:
26135:
26134:
26129:
26126:
26124:
26121:
26118:
26115:
26112:
26108:
26103:
26102:
26093:
26087:
26083:
26078:
26073:
26072:
26066:
26063:
26061:0-7679-0816-3
26057:
26053:
26049:
26048:
26043:
26039:
26035:
26029:
26025:
26020:
26017:
26011:
26007:
26006:
26002:(July 1998),
26001:
25997:
25993:
25989:
25983:
25979:
25975:
25971:
25967:
25961:
25957:
25952:
25948:
25942:
25938:
25934:
25929:
25928:
25916:
25911:
25904:
25899:
25892:
25888:
25883:
25878:
25874:
25870:
25867:(3): 857â78,
25866:
25862:
25855:
25848:
25844:
25840:
25836:
25832:
25828:
25824:
25823:
25815:
25808:
25807:
25799:
25792:
25787:
25774:on 2020-09-25
25770:
25766:
25759:
25752:
25750:
25740:
25732:
25728:
25721:
25715:
25708:
25702:
25698:
25694:
25693:
25688:
25684:
25680:
25674:
25667:
25662:
25655:
25651:
25647:
25643:
25636:
25629:
25623:
25619:
25615:
25608:
25601:
25590:
25588:9782705678128
25584:
25580:
25579:
25571:
25564:
25559:
25552:
25547:
25539:
25532:
25531:
25523:
25515:
25511:
25505:
25497:
25496:
25488:
25481:
25475:
25471:
25464:
25457:on 2006-05-26
25453:
25449:
25445:
25442:(3): 255â74,
25441:
25437:
25430:
25423:
25409:
25408:
25400:
25393:
25392:
25384:
25376:
25370:
25362:
25355:
25348:
25344:
25339:
25335:
25334:
25326:
25319:
25313:
25309:
25305:
25299:
25292:
25287:
25283:
25279:
25273:
25269:
25268:
25260:
25253:
25249:
25245:
25241:
25234:
25226:
25222:
25218:
25216:
25207:
25200:
25194:
25190:
25183:
25176:
25171:
25164:
25160:
25156:
25152:
25148:
25133:
25125:
25124:
25118:
25114:
25108:
25101:
25097:
25093:
25089:
25086:(3): 278â79,
25085:
25081:
25074:
25065:
25057:
25053:
25046:
25045:
25038:
25029:
25022:
25018:
25011:
25010:
25003:
24996:
24991:
24984:
24979:
24972:
24967:
24960:
24954:
24950:
24946:
24940:
24932:
24931:
24925:
24921:
24915:
24907:
24906:
24898:
24891:
24887:
24883:
24879:
24875:
24871:
24867:
24860:
24853:
24849:
24845:
24841:
24837:
24830:
24823:
24819:
24815:
24811:
24807:
24803:
24799:
24795:
24792:(2): 313â18,
24791:
24787:
24780:
24772:
24768:
24761:
24754:
24747:
24743:
24739:
24735:
24731:
24727:
24722:
24717:
24713:
24709:
24702:
24694:
24690:
24683:
24676:
24672:
24668:
24664:
24660:
24656:
24649:
24642:
24636:
24632:
24625:
24612:on 2023-11-18
24608:
24604:
24600:
24596:
24590:
24586:
24582:
24581:Butler, Steve
24575:
24571:
24565:
24557:
24556:
24550:
24546:
24540:
24533:
24532:Prime Numbers
24528:
24520:
24516:
24511:
24506:
24501:
24496:
24493:(3): 366â70,
24492:
24488:
24487:
24470:
24466:
24460:
24457:
24454:
24450:
24438:
24425:on 2009-12-14
24424:
24420:
24419:
24411:
24403:
24399:
24393:
24386:
24384:9781470457181
24380:
24376:
24372:
24365:
24358:
24354:
24350:
24346:
24345:
24337:
24329:
24328:
24322:
24314:
24308:
24301:
24297:
24292:
24285:
24281:
24277:
24273:
24269:
24265:
24261:
24257:
24250:
24242:
24241:
24236:
24232:
24225:
24223:
24221:
24213:
24207:
24203:
24202:
24194:
24187:
24181:
24177:
24170:
24163:
24158:
24148:
24147:
24142:
24136:
24122:
24118:
24112:
24097:
24093:
24092:
24084:
24077:
24069:
24063:
24056:
24050:
24046:
24039:
24031:
24030:
24024:
24020:
24014:
24006:
24005:
23999:
23995:
23989:
23982:
23977:
23970:
23965:
23958:
23952:
23948:
23944:
23940:
23933:
23927:
23922:
23916:
23912:
23911:
23903:
23896:
23891:
23885:
23881:
23877:
23871:
23863:
23859:
23858:
23850:
23836:
23835:
23830:
23823:
23816:
23814:1-4027-3522-7
23810:
23806:
23799:
23784:
23783:
23778:
23772:
23765:
23760:
23745:
23744:
23736:
23729:
23722:
23717:
23709:
23702:
23694:
23689:
23685:
23681:
23677:
23676:
23668:
23661:
23656:
23652:
23646:
23639:
23634:
23632:
23624:
23619:
23613:
23609:
23608:
23603:
23602:Knuth, Donald
23597:
23590:
23585:
23578:
23573:
23566:
23561:
23554:
23549:
23542:
23538:
23531:
23525:
23521:
23520:
23515:
23514:Knuth, Donald
23509:
23507:
23498:
23493:
23490:(3): 229â44,
23489:
23485:
23478:
23476:
23468:
23462:
23458:
23457:
23449:
23447:
23445:
23436:
23435:
23429:
23425:
23419:
23417:
23412:
23398:
23392:
23388:
23372:
23371:Wythoff array
23369:
23366:
23363:
23360:
23357:
23355:
23352:
23349:
23346:
23345:
23336:
23332:
23328:
23325:
23322:
23319:
23316:
23312:
23308:
23304:
23298:
23294:
23290:
23285:
23281:
23278:
23274:
23271:
23266:
23255:
23246:
23242:
23238:
23237:
23231:
23229:
23211:
23208:
23203:
23199:
23174:
23171:
23166:
23162:
23137:
23134:
23129:
23125:
23100:
23097:
23092:
23088:
23065:
23062:
23057:
23053:
23044:
23040:
23030:
23026:
23021:
23017:
23009:
23005:
22998:
22994:
22984:
22981:
22977:
22976:
22975:
22973:
22972:haplodiploids
22969:
22964:
22952:
22948:
22944:
22940:are those at
22933:
22929:
22925:
22921:
22916:
22912:
22908:
22895:
22880:
22875:
22872:
22869:
22863:
22860:
22853:
22849:
22844:
22841:
22835:
22832:
22824:
22820:
22815:
22811:
22807:
22798:
22792:
22788:
22786:
22782:
22778:
22774:
22772:
22768:
22764:
22760:
22756:
22752:
22748:
22744:
22740:
22736:
22732:
22728:
22724:
22716:
22712:
22707:
22701:
22689:
22686:
22684:
22681:
22678:
22674:
22670:
22667:
22663:
22659:
22655:
22652:
22648:
22644:
22641:
22637:
22634:
22630:
22627:
22623:
22620:
22616:
22612:
22608:
22605:
22604:
22599:
22590:
22586:
22583:
22579:
22575:
22571:
22570:
22564:
22559:
22547:
22543:
22539:
22535:
22532:
22528:
22524:
22520:
22516:
22500:
22495:
22488:
22485:
22482:
22479:
22475:
22469:
22464:
22454:
22451:
22448:
22444:
22438:
22435:
22432:
22428:
22424:
22418:
22413:
22403:
22400:
22397:
22393:
22387:
22383:
22371:
22367:
22363:
22359:
22356:
22352:
22348:
22344:
22341:
22337:
22334:
22331:
22327:
22323:
22320:
22304:
22296:
22292:
22274:
22264:
22260:
22253:
22234:
22226:
22222:
22209:
22204:
22199:
22190:
22186:
22182:
22170:
22157:
22153:
22148:
22135:
22127:
22123:
22110:
22109:
22108:
22106:
22102:
22099:
22094:
22092:
22077:
22072:
22067:
22065:
22064:
22058:
22053:
22048:
22043:
22038:
22033:
22032:
22029:
22028:
22027:
22020:
22013:
21996:
21993:
21990:
21982:
21975:
21971:
21967:); there are
21966:
21949:
21945:
21941:
21937:
21907:
21904:
21893:
21882:
21879:
21868:
21857:
21854:
21827:
21822:
21817:
21815:
21814:
21808:
21803:
21798:
21793:
21791:
21790:
21784:
21779:
21778:
21775:
21774:
21773:
21770:
21754:
21750:
21727:
21723:
21717:
21713:
21702:
21699:
21696:
21688:
21685:
21682:
21677:
21669:
21666:
21663:
21655:
21652:
21649:
21645:
21641:
21638:
21635:
21630:
21622:
21619:
21616:
21608:
21604:
21600:
21594:
21591:
21588:
21580:
21576:
21572:
21569:
21566:
21558:
21554:
21550:
21547:
21544:
21541:
21537:
21515:
21504:
21500:
21497:
21494:
21491:
21488:
21473:
21468:
21464:
21461:
21458:
21452:
21446:
21443:
21440:
21436:
21432:
21427:
21423:
21414:
21410:
21402:
21397:
21380:
21376:
21371:
21367:
21363:
21359:
21355:
21351:
21346:
21342:
21338:
21335:
21324:
21320:
21314:
21310:
21306:
21298:
21294:
21291:
21284:
21280:
21274:
21270:
21266:
21253:
21243:
21238:
21237:Lucas numbers
21234:
21231:
21227:
21224:
21223:negafibonacci
21220:
21219:
21218:
21215:
21213:
21209:
21205:
21199:
21189:
21187:
21179:
21175:
21171:
21167:
21163:
21154:
21148:
21138:
21137:Pisano period
21131:
21125:
21121:
21114:
21110:
21104:
21091:
21088:
21085:
21082:
21079:
21076:
21071:
21066:
21061:
21057:
21050:
21047:
21044:
21039:
21035:
21028:
21025:
21022:
21017:
21013:
21006:
21003:
21000:
20997:
20994:
20989:
20984:
20979:
20975:
20968:
20965:
20962:
20957:
20953:
20946:
20943:
20940:
20935:
20931:
20924:
20921:
20918:
20913:
20909:
20902:
20899:
20896:
20891:
20887:
20879:For example,
20877:
20873:
20869:
20862:
20858:
20848:
20831:
20827:
20819:
20816:
20813:
20810:
20805:
20798:
20794:
20788:
20772:
20768:
20760:
20757:
20754:
20751:
20746:
20739:
20735:
20729:
20710:
20707:
20702:
20698:
20689:
20686:
20681:
20677:
20657:
20654:
20650:
20646:
20643:
20630:
20627:
20614:
20610:
20603:
20600:
20593:
20590:
20587:
20583:
20579:
20576:
20563:
20560:
20547:
20543:
20536:
20533:
20526:
20523:
20520:
20517:
20504:
20501:
20484:and we have:
20481:
20474:
20470:
20466:
20449:
20445:
20437:
20434:
20431:
20428:
20423:
20416:
20412:
20406:
20390:
20386:
20378:
20375:
20372:
20369:
20366:
20361:
20354:
20350:
20344:
20325:
20322:
20317:
20313:
20304:
20301:
20296:
20292:
20272:
20269:
20265:
20261:
20258:
20245:
20242:
20229:
20225:
20218:
20215:
20208:
20205:
20202:
20199:
20195:
20191:
20188:
20175:
20172:
20159:
20155:
20148:
20145:
20138:
20135:
20132:
20129:
20116:
20113:
20096:and we have:
20093:
20086:
20082:
20078:
20061:
20057:
20049:
20046:
20043:
20040:
20035:
20028:
20024:
20018:
20002:
19998:
19990:
19987:
19984:
19981:
19976:
19969:
19965:
19959:
19940:
19937:
19932:
19928:
19919:
19916:
19911:
19907:
19887:
19884:
19880:
19876:
19873:
19860:
19857:
19844:
19840:
19833:
19830:
19823:
19820:
19817:
19813:
19809:
19806:
19793:
19790:
19777:
19773:
19766:
19763:
19756:
19753:
19750:
19747:
19734:
19731:
19714:and we have:
19711:
19704:
19700:
19696:
19679:
19675:
19667:
19664:
19661:
19658:
19655:
19650:
19643:
19639:
19633:
19617:
19613:
19605:
19602:
19599:
19596:
19593:
19588:
19581:
19577:
19571:
19552:
19549:
19544:
19540:
19531:
19528:
19523:
19519:
19499:
19496:
19493:
19489:
19485:
19482:
19469:
19466:
19453:
19449:
19442:
19439:
19432:
19429:
19426:
19423:
19419:
19415:
19412:
19399:
19396:
19383:
19379:
19372:
19369:
19362:
19359:
19356:
19353:
19340:
19337:
19320:and we have:
19317:
19310:
19306:
19302:
19282:
19275:
19271:
19263:
19260:
19257:
19243:
19239:
19230:
19226:
19223:
19210:
19207:
19194:
19190:
19183:
19180:
19166:
19162:
19154:
19151:
19148:
19134:
19130:
19121:
19117:
19114:
19101:
19098:
19085:
19081:
19074:
19071:
19062:
19057:
19052:
19044:
19040:
19037:
19034:
19028:
19022:
19012:
19006:
19004:
18999:
18986:
18977:
18973:
18968:
18960:
18957:
18951:
18946:
18943:
18938:
18934:
18931:
18927:
18918:
18911:
18894:
18891:
18889:
18882:
18878:
18872:
18869:
18866:
18864:
18857:
18853:
18847:
18844:
18841:
18838:
18836:
18821:
18818:
18801:
18798:
18795:
18793:
18786:
18782:
18776:
18773:
18770:
18768:
18761:
18757:
18751:
18748:
18745:
18742:
18740:
18725:
18722:
18705:
18702:
18699:
18697:
18690:
18686:
18680:
18677:
18674:
18672:
18657:
18654:
18637:
18634:
18631:
18629:
18622:
18618:
18612:
18609:
18606:
18604:
18597:
18593:
18587:
18584:
18581:
18578:
18576:
18561:
18558:
18541:
18538:
18535:
18533:
18526:
18522:
18516:
18513:
18510:
18508:
18501:
18497:
18491:
18488:
18485:
18482:
18480:
18465:
18462:
18441:For example,
18439:
18426:
18419:
18415:
18407:
18404:
18398:
18393:
18390:
18385:
18381:
18378:
18374:
18359:
18355:
18346:
18341:
18338:
18333:
18329:
18324:
18320:
18306:
18286:
18279:
18275:
18267:
18264:
18261:
18258:
18248:
18245:
18234:
18230:
18222:
18219:
18216:
18213:
18203:
18196:
18193:
18190:
18180:
18174:
18169:
18165:
18160:
18157:
18152:
18121:
18118:
18101:
18092:
18086:
18077:
18072:
18064:
18055:
18046:
18039:
18028:
18026:
18022:
18017:
18015:
18011:
18007:
18002:
18000:
17999:perfect power
17996:
17991:
17984:
17978:
17976:
17972:
17963:
17956:
17952:
17945:
17941:
17936:
17930:
17929:
17928:
17926:
17922:
17916:
17906:
17904:
17900:
17895:
17878:
17871:
17867:
17857:
17851:
17845:
17840:
17833:
17828:
17822:
17816:
17811:
17803:
17800:
17797:
17793:
17785:
17781:
17771:
17767:
17759:
17756:
17753:
17749:
17742:
17732:
17728:
17723:
17719:
17714:
17706:
17698:
17682:
17681:Jacobi symbol
17666:
17660:
17655:
17652:
17647:
17642:
17638:
17634:
17630:
17627:
17619:
17609:
17596:
17590:
17585:
17582:
17577:
17572:
17568:
17564:
17560:
17557:
17549:
17545:
17540:
17520:
17515:
17512:
17509:
17505:
17501:
17498:
17486:
17482:
17474:
17471:
17468:
17465:
17458:
17453:
17450:
17447:
17443:
17439:
17436:
17424:
17420:
17412:
17409:
17406:
17403:
17396:
17391:
17387:
17383:
17380:
17372:
17369:
17366:
17360:
17350:
17348:
17335:
17327:
17323:
17307:
17303:
17290:
17280:
17275:
17254:
17251:
17243:
17240:
17237:
17233:
17229:
17224:
17221:
17218:
17214:
17204:
17196:
17193:
17190:
17186:
17182:
17177:
17173:
17163:
17155:
17152:
17149:
17145:
17141:
17136:
17132:
17118:
17117:
17116:
17102:
17099:
17094:
17090:
17069:
17066:
17061:
17057:
17049:because both
17048:
17043:
17041:
17015:
17012:
17009:
17006:
17003:
17000:
16997:
16987:
16983:
16977:
16974:
16969:
16965:
16961:
16956:
16952:
16948:
16943:
16939:
16924:
16920:
16897:
16894:
16889:
16885:
16865:
16852:
16843:
16839:
16834:
16828:
16825:
16820:
16816:
16811:
16805:
16802:
16799:
16790:
16786:
16781:
16777:
16770:
16765:
16762:
16759:
16755:
16730:
16725:
16719:
16714:
16711:
16705:
16696:
16692:
16687:
16683:
16671:
16668:
16665:
16661:
16652:
16648:
16646:
16642:
16637:
16624:
16621:
16618:
16611:
16608:
16604:
16600:
16588:
16585:
16582:
16578:
16574:
16567:
16564:
16561:
16558:
16554:
16550:
16538:
16535:
16532:
16528:
16524:
16517:
16513:
16509:
16497:
16494:
16491:
16487:
16478:
16473:
16459:
16454:
16445:
16442:
16438:
16434:
16431:
16425:
16422:
16418:
16412:
16404:
16401:
16397:
16393:
16390:
16384:
16381:
16377:
16370:
16364:
16359:
16352:
16349:
16345:
16341:
16317:
16309:
16305:
16301:
16298:
16292:
16288:
16275:
16272:
16269:
16265:
16261:
16255:
16249:
16241:
16224:
16215:
16211:
16204:
16201:
16193:
16189:
16182:
16178:
16172:
16167:
16160:
16157:
16153:
16149:
16137:
16134:
16131:
16127:
16117:
16104:
16099:
16095:
16092:
16087:
16079:
16071:
16064:
16060:
16052:
16047:
16044:
16041:
16037:
16029:
16026:
16023:
16015:
16012:
15996:
15993:
15990:
15986:
15977:
15973:
15968:
15955:
15950:
15946:
15940:
15932:
15929:
15926:
15923:
15919:
15915:
15912:
15908:
15896:
15893:
15890:
15886:
15876:
15863:
15859:
15855:
15852:
15847:
15842:
15836:
15830:
15825:
15822:
15816:
15813:
15809:
15801:
15797:
15793:
15788:
15783:
15777:
15771:
15766:
15763:
15757:
15754:
15750:
15742:
15738:
15733:
15726:
15723:
15718:
15711:
15704:
15700:
15694:
15682:
15679:
15676:
15672:
15662:
15649:
15644:
15639:
15633:
15627:
15622:
15619:
15613:
15610:
15606:
15598:
15594:
15587:
15583:
15577:
15570:
15567:
15564:
15561:
15557:
15553:
15541:
15538:
15535:
15531:
15522:
15518:
15508:
15495:
15492:
15489:
15486:
15481:
15478:
15473:
15468:
15465:
15460:
15454:
15448:
15425:
15419:
15399:
15390:
15377:
15373:
15367:
15364:
15359:
15355:
15350:
15347:
15341:
15335:
15327:
15326:
15307:
15301:
15294:numbers, and
15293:
15292:negafibonacci
15276:
15272:
15268:
15264:
15261:
15257:
15253:
15250:
15244:
15235:
15233:
15216:
15210:
15205:
15202:
15198:
15191:
15188:
15182:
15179:
15158:
15152:
15147:
15144:
15140:
15133:
15130:
15124:
15121:
15100:
15093:
15090:
15087:
15084:
15080:
15075:
15069:
15066:
15063:
15060:
15056:
15050:
15043:
15039:
15034:
15028:
15022:
15014:
15009:
14995:
14992:
14989:
14967:
14963:
14953:
14936:
14933:
14930:
14928:
14918:
14914:
14905:
14902:
14899:
14895:
14891:
14886:
14883:
14880:
14876:
14872:
14867:
14863:
14849:
14846:
14843:
14839:
14835:
14830:
14826:
14822:
14819:
14814:
14810:
14806:
14803:
14798:
14794:
14790:
14787:
14785:
14775:
14771:
14765:
14762:
14759:
14755:
14744:
14741:
14738:
14734:
14730:
14725:
14721:
14715:
14712:
14709:
14705:
14694:
14691:
14688:
14684:
14680:
14675:
14671:
14665:
14661:
14650:
14647:
14644:
14640:
14636:
14634:
14624:
14621:
14618:
14614:
14608:
14604:
14593:
14590:
14587:
14583:
14579:
14574:
14571:
14568:
14564:
14558:
14554:
14543:
14540:
14537:
14533:
14529:
14524:
14520:
14514:
14510:
14499:
14496:
14493:
14489:
14485:
14483:
14475:
14469:
14461:
14457:
14453:
14450:
14447:
14444:
14412:
14408:
14404:
14401:
14398:
14395:
14383:
14370:
14362:
14358:
14354:
14351:
14348:
14345:
14341:
14336:
14330:
14324:
14316:
14302:
14299:
14295:
14291:
14288:
14280:
14255:
14248:
14243:
14230:
14227:
14224:
14219:
14215:
14211:
14208:
14203:
14199:
14195:
14192:
14187:
14183:
14179:
14176:
14171:
14167:
14163:
14160:
14157:
14154:
14151:
14146:
14142:
14136:
14132:
14121:
14118:
14115:
14111:
14107:
14101:
14095:
14087:
14086:
14082:
14072:
14070:
14064:
14058:
14045:
14040:
14037:
14034:
14027:
14024:
14021:
14017:
14009:
14002:
13998:
13990:
13987:
13984:
13980:
13968:
13965:
13952:
13947:
13944:
13941:
13937:
13933:
13928:
13925:
13922:
13919:
13915:
13906:
13903:
13890:
13885:
13882:
13879:
13872:
13869:
13866:
13862:
13854:
13847:
13843:
13835:
13832:
13829:
13825:
13813:
13810:
13797:
13792:
13789:
13786:
13782:
13778:
13773:
13770:
13767:
13764:
13760:
13751:
13748:
13746:
13742:
13738:
13721:
13715:
13708:
13705:
13702:
13698:
13692:
13689:
13684:
13677:
13673:
13666:
13660:
13653:
13649:
13643:
13640:
13636:
13630:
13623:
13619:
13613:
13610:
13605:
13598:
13595:
13592:
13588:
13581:
13575:
13572:
13569:
13565:
13559:
13555:
13551:
13548:
13543:
13540:
13536:
13514:
13507:
13503:
13497:
13492:
13488:
13482:
13475:
13472:
13469:
13465:
13459:
13456:
13451:
13444:
13441:
13438:
13434:
13428:
13423:
13420:
13417:
13414:
13410:
13388:
13381:
13377:
13371:
13366:
13359:
13355:
13347:
13344:
13341:
13337:
13333:
13330:
13325:
13318:
13315:
13312:
13308:
13302:
13297:
13294:
13291:
13288:
13284:
13275:
13259:
13255:
13249:
13241:
13238:
13232:
13229:
13224:
13217:
13213:
13207:
13204:
13199:
13196:
13193:
13189:
13183:
13180:
13177:
13173:
13167:
13163:
13159:
13156:
13151:
13144:
13140:
13134:
13131:
13126:
13123:
13119:
13106:
13096:
13092:
13073:
13069:
13063:
13059:
13055:
13051:
13045:
13042:
13039:
13035:
13031:
13026:
13023:
13020:
13016:
13011:
13005:
13001:
12997:
12992:
12985:
12982:
12979:
12975:
12969:
12964:
12957:
12954:
12951:
12947:
12941:
12936:
12933:
12929:
12907:
12904:
12901:
12897:
12891:
12883:
12880:
12874:
12869:
12865:
12859:
12856:
12853:
12849:
12845:
12840:
12837:
12834:
12830:
12824:
12820:
12806:
12791:
12784:
12780:
12772:
12769:
12766:
12758:
12755:
12749:
12744:
12741:
12738:
12734:
12728:
12725:
12722:
12718:
12714:
12709:
12702:
12698:
12673:
12670:
12667:
12659:
12656:
12650:
12645:
12642:
12639:
12635:
12629:
12626:
12623:
12619:
12615:
12610:
12603:
12599:
12586:
12576:
12568:
12552:
12547:
12544:
12541:
12538:
12518:
12515:
12512:
12509:
12506:
12479:
12476:
12471:
12468:
12465:
12461:
12452:
12447:
12445:
12434:
12431:
12428:
12422:
12417:
12414:
12411:
12407:
12403:
12398:
12395:
12392:
12388:
12384:
12382:
12371:
12368:
12363:
12360:
12355:
12352:
12349:
12345:
12341:
12338:
12335:
12330:
12327:
12324:
12320:
12316:
12310:
12308:
12297:
12294:
12289:
12286:
12281:
12278:
12275:
12271:
12264:
12258:
12255:
12250:
12247:
12242:
12239:
12236:
12232:
12225:
12223:
12212:
12209:
12206:
12201:
12198:
12193:
12190:
12187:
12183:
12176:
12170:
12167:
12164:
12159:
12156:
12151:
12148:
12145:
12141:
12134:
12132:
12123:
12117:
12113:
12109:
12106:
12103:
12098:
12094:
12090:
12087:
12084:
12081:
12077:
12073:
12068:
12064:
12060:
12057:
12054:
12049:
12045:
12041:
12038:
12033:
12030:
12004:
11982:
11979:
11974:
11971:
11968:
11964:
11960:
11955:
11951:
11945:
11940:
11937:
11934:
11930:
11920:
11906:
11901:
11897:
11893:
11888:
11884:
11880:
11875:
11871:
11865:
11849:
11834:
11831:
11828:
11824:
11818:
11815:
11812:
11808:
11804:
11799:
11794:
11790:
11784:
11781:
11778:
11773:
11770:
11767:
11763:
11742:
11736:
11733:
11730:
11726:
11722:
11717:
11713:
11708:
11702:
11699:
11696:
11692:
11688:
11683:
11676:
11673:
11670:
11666:
11660:
11655:
11650:
11646:
11640:
11635:
11632:
11629:
11625:
11602:
11599:
11596:
11592:
11586:
11582:
11578:
11573:
11568:
11564:
11558:
11553:
11550:
11547:
11543:
11520:
11513:
11510:
11507:
11503:
11491:
11478:
11475:
11470:
11467:
11464:
11460:
11456:
11451:
11447:
11441:
11438:
11435:
11430:
11427:
11424:
11420:
11400:
11397:
11394:
11371:
11368:
11363:
11360:
11357:
11353:
11349:
11344:
11341:
11338:
11334:
11330:
11325:
11322:
11319:
11315:
11311:
11306:
11302:
11296:
11291:
11288:
11285:
11281:
11273:
11272:
11271:
11255:
11252:
11249:
11245:
11224:
11221:
11216:
11213:
11210:
11206:
11202:
11197:
11193:
11187:
11182:
11179:
11176:
11172:
11162:
11160:
11150:
11137:
11129:
11125:
11121:
11118:
11115:
11112:
11108:
11099:
11081:
11077:
11071:
11067:
11056:
11053:
11050:
11046:
11037:
11032:
11018:
11015:
11012:
11004:
11000:
10984:
10938:
10935:
10927:
10923:
10900:
10897:
10887:
10883:
10872:The sequence
10865:
10863:
10858:
10856:
10838:
10834:
10830:
10827:
10824:
10821:
10818:
10813:
10810:
10807:
10803:
10799:
10794:
10790:
10767:
10764:
10761:
10757:
10753:
10748:
10744:
10733:
10709:
10705:
10682:
10679:
10676:
10672:
10666:
10662:
10658:
10653:
10648:
10644:
10638:
10633:
10630:
10627:
10623:
10613:
10609:
10589:
10586:
10582:
10573:
10567:
10547:
10544:
10541:
10538:
10534:
10525:
10509:
10506:
10501:
10498:
10495:
10492:
10488:
10484:
10479:
10476:
10472:
10466:
10461:
10458:
10455:
10451:
10428:
10425:
10421:
10417:
10412:
10409:
10406:
10403:
10399:
10393:
10390:
10387:
10382:
10379:
10376:
10372:
10362:
10348:
10345:
10340:
10337:
10334:
10330:
10326:
10321:
10317:
10311:
10306:
10303:
10300:
10296:
10275:
10272:
10267:
10263:
10228:
10225:
10222:
10219:
10216:
10213:
10210:
10207:
10204:
10190:
10176:
10173:
10170:
10167:
10164:
10161:
10158:
10155:
10152:
10149:
10146:
10132:
10129:
10126:
10123:
10120:
10115:
10112:
10109:
10105:
10101:
10096:
10092:
10088:
10083:
10080:
10077:
10073:
10065:
10064:
10063:
10060:
10040:
10037:
10034:
10031:
10028:
10025:
10022:
10019:
10016:
10007:
9998:
9995:
9992:
9989:
9986:
9983:
9980:
9977:
9974:
9971:
9968:
9942:
9939:
9936:
9933:
9930:
9924:
9921:
9918:
9915:
9912:
9909:
9906:
9900:
9894:
9891:
9888:
9885:
9882:
9859:
9856:
9853:
9844:
9831:
9828:
9823:
9820:
9817:
9813:
9809:
9804:
9800:
9794:
9789:
9786:
9783:
9779:
9768:
9757:
9741:
9737:
9714:
9711:
9708:
9704:
9700:
9695:
9692:
9689:
9685:
9662:
9659:
9656:
9652:
9629:
9626:
9623:
9619:
9598:
9595:
9592:
9572:
9569:
9566:
9538:
9535:
9532:
9529:
9523:
9520:
9517:
9514:
9511:
9505:
9499:
9496:
9493:
9490:
9487:
9473:
9462:
9459:
9456:
9453:
9447:
9444:
9441:
9438:
9435:
9429:
9423:
9420:
9417:
9414:
9411:
9397:
9392:
9388:
9365:
9361:
9338:
9335:
9332:
9328:
9324:
9319:
9316:
9313:
9309:
9305:
9300:
9296:
9261:
9258:
9255:
9249:
9243:
9240:
9237:
9234:
9231:
9225:
9219:
9216:
9213:
9210:
9207:
9201:
9195:
9192:
9189:
9186:
9183:
9177:
9171:
9168:
9165:
9162:
9159:
9156:
9153:
9139:
9136:
9133:
9128:
9124:
9116:
9091:
9088:
9085:
9079:
9073:
9070:
9067:
9061:
9055:
9052:
9049:
9046:
9043:
9029:
9026:
9023:
9018:
9014:
9006:
8981:
8975:
8969:
8966:
8963:
8949:
8946:
8943:
8938:
8934:
8926:
8901:
8887:
8884:
8881:
8876:
8872:
8864:
8828:
8825:
8822:
8817:
8813:
8805:
8775:
8772:
8769:
8764:
8760:
8752:
8751:
8750:
8748:
8744:
8722:
8719:
8716:
8690:
8687:
8682:
8678:
8657:
8654:
8649:
8645:
8622:
8618:
8597:
8594:
8591:
8569:
8565:
8556:
8541:
8539:
8529:
8525:
8520:
8515:
8498:
8493:
8489:
8480:
8476:
8472:
8467:
8464:
8461:
8457:
8453:
8447:
8445:
8435:
8431:
8422:
8418:
8414:
8409:
8406:
8403:
8399:
8395:
8389:
8387:
8377:
8373:
8364:
8361:
8358:
8354:
8350:
8345:
8342:
8339:
8335:
8328:
8326:
8316:
8313:
8305:
8302:
8298:
8288:
8281:
8278:
8275:
8271:
8265:
8260:
8253:
8249:
8243:
8241:
8234:
8231:
8228:
8225:
8221:
8207:
8203:
8197:
8180:
8175:
8172:
8169:
8165:
8161:
8159:
8152:
8148:
8142:
8139:
8136:
8132:
8128:
8123:
8120:
8117:
8113:
8107:
8103:
8095:
8090:
8087:
8084:
8081:
8078:
8074:
8070:
8068:
8060:
8057:
8054:
8050:
8042:
8039:
8036:
8032:
8027:
8021:
8017:
8009:
8005:
7989:
7980:
7976:
7971:
7967:
7966:square matrix
7962:
7958:
7955:
7949:
7936:
7931:
7924:
7920:
7914:
7909:
7906:
7903:
7899:
7893:
7890:
7887:
7883:
7879:
7874:
7866:
7863:
7852:
7847:
7845:
7839:
7835:
7825:
7812:
7807:
7799:
7796:
7793:
7789:
7781:
7777:
7767:
7763:
7755:
7752:
7749:
7745:
7738:
7733:
7728:
7722:
7716:
7711:
7704:
7699:
7693:
7682:
7678:
7674:
7668:
7664:
7657:
7645:
7641:
7635:
7631:
7626:
7622:
7613:
7608:
7595:
7577:
7574:
7571:
7559:
7548:
7545:
7533:
7522:
7519:
7507:
7496:
7493:
7490:
7487:
7479:
7473:
7468:
7466:
7462:
7457:
7451:
7438:
7429:
7414:
7411:
7403:
7400:
7394:
7389:
7385:
7373:
7368:
7364:
7354:
7337:
7326:
7323:
7310:
7304:
7299:
7296:
7287:
7284:
7280:
7274:
7268:
7260:
7256:
7249:
7241:
7238:
7230:
7227:
7219:
7212:
7205:
7201:
7194:
7187:
7181:
7176:
7167:
7164:
7160:
7156:
7151:
7145:
7140:
7138:
7120:
7116:
7110:
7106:
7092:
7089:
7085:
7079:
7071:
7068:
7060:
7057:
7049:
7042:
7035:
7031:
7024:
7019:
7016:
7014:
6996:
6992:
6986:
6982:
6968:
6965:
6961:
6955:
6947:
6944:
6942:
6924:
6920:
6914:
6910:
6896:
6892:
6888:
6886:
6871:
6867:
6861:
6858:
6855:
6851:
6830:
6823:
6810:
6805:
6799:
6794:
6785:
6782:
6778:
6774:
6769:
6763:
6758:
6755:
6751:
6746:
6737:
6734:
6730:
6726:
6721:
6714:
6709:
6703:
6698:
6687:
6684:
6667:
6662:
6659:
6655:
6649:
6641:
6638:
6636:
6629:
6625:
6617:
6612:
6609:
6605:
6598:
6595:
6593:
6588:
6576:
6574:
6569:
6564:
6559:
6546:
6541:
6535:
6527:
6513:
6508:
6505:
6494:
6483:
6470:
6458:
6453:
6447:
6439:
6425:
6420:
6417:
6406:
6395:
6382:
6371:
6366:
6362:
6353:
6344:
6327:
6316:
6310:
6307:
6303:
6299:
6285:
6279:
6271:
6257:
6252:
6249:
6238:
6227:
6214:
6202:
6190:
6187:
6174:
6168:
6160:
6146:
6141:
6138:
6127:
6116:
6103:
6092:
6090:
6076:
6068:
6065:
6057:
6054:
6045:
6041:
6036:
6027:
6019:
6015:
6008:
6004:
5999:
5997:
5983:
5975:
5971:
5964:
5960:
5955:
5946:
5938:
5934:
5927:
5923:
5918:
5916:
5909:
5899:
5878:
5865:
5856:
5847:
5843:
5838:
5829:
5820:
5816:
5811:
5800:
5797:
5786:
5781:
5771:
5758:
5745:
5734:
5728:
5725:
5721:
5717:
5705:
5696:
5689:
5678:
5675:
5664:
5655:
5645:
5616:
5611:
5608:
5595:
5592:
5586:
5581:
5578:
5574:
5570:
5567:
5564:
5531:
5526:
5523:
5510:
5507:
5501:
5498:
5489:
5485:
5481:
5463:
5453:
5444:
5434:
5429:
5419:
5408:which yields
5406:
5393:
5388:
5378:
5366:
5361:
5358:
5355:
5345:
5311:
5307:
5301:
5298:
5295:
5291:
5277:
5271:
5266:
5259:
5254:
5248:
5243:
5230:
5227:
5224:
5220:
5214:
5211:
5208:
5204:
5187:
5185:
5175:
5158:
5152:
5142:
5139:
5136:
5132:
5128:
5125:
5120:
5116:
5109:
5104:
5096:
5093:
5087:
5084:
5079:
5072:
5068:
5062:
5055:
5054:
5053:
5039:
5035:
5026:
5023:
5020:
5016:
5012:
5009:
5004:
5000:
4996:
4991:
4984:
4980:
4973:
4968:
4965:
4962:
4958:
4954:
4951:
4946:
4942:
4938:
4933:
4929:
4919:
4906:
4901:
4893:
4885:
4882:
4876:
4873:
4868:
4861:
4857:
4851:
4846:
4841:
4834:
4830:
4823:
4818:
4814:
4805:
4803:
4785:
4781:
4772:
4754:
4750:
4744:
4720:
4714:
4705:
4702:
4698:
4692:
4684:
4681:
4675:
4670:
4666:
4659:
4654:
4650:
4641:
4625:
4622:
4617:
4613:
4609:
4589:
4586:
4581:
4577:
4573:
4565:
4551:
4538:
4533:
4530:
4527:
4523:
4519:
4514:
4511:
4508:
4504:
4500:
4495:
4491:
4482:
4478:
4469:
4463:
4450:
4445:
4442:
4439:
4435:
4431:
4428:
4423:
4419:
4415:
4410:
4406:
4385:
4382:
4379:
4376:
4371:
4367:
4346:
4342:
4338:
4335:
4332:
4329:
4309:
4304:
4300:
4296:
4293:
4288:
4285:
4282:
4278:
4274:
4269:
4265:
4261:
4258:
4250:
4247:
4244:
4240:
4236:
4231:
4227:
4220:
4217:
4212:
4209:
4206:
4202:
4198:
4192:
4189:
4186:
4178:
4174:
4170:
4167:
4162:
4159:
4156:
4152:
4148:
4143:
4139:
4133:
4129:
4125:
4122:
4114:
4111:
4108:
4104:
4100:
4097:
4092:
4088:
4081:
4076:
4073:
4070:
4066:
4055:
4050:
4046:
4030:
4025:
4022:
4019:
4015:
4011:
4008:
4003:
3999:
3995:
3990:
3986:
3977:
3973:
3957:
3935:
3931:
3921:
3908:
3905:
3902:
3899:
3896:
3891:
3887:
3866:
3862:
3861:
3860:
3846:
3824:
3820:
3816:
3809:
3805:
3799:
3796:
3793:
3789:
3775:
3761:
3759:
3743:
3739:
3733:
3729:
3725:
3722:
3717:
3713:
3690:
3686:
3663:
3659:
3649:
3636:
3633:
3630:
3623:
3619:
3613:
3610:
3607:
3603:
3589:
3565:
3562:
3554:
3550:
3541:
3528:
3522:
3519:
3516:
3511:
3508:
3505:
3502:
3496:
3493:
3490:
3485:
3481:
3477:
3467:
3463:
3455:
3450:
3442:
3425:
3421:
3418:
3415:
3412:
3407:
3403:
3399:
3389:
3385:
3366:
3360:
3354:
3350:
3341:
3337:
3324:
3310:
3307:
3304:
3298:
3292:
3287:
3283:
3262:
3259:
3256:
3250:
3244:
3241:
3218:
3212:
3207:
3203:
3198:
3191:
3185:
3180:
3176:
3172:
3166:
3160:
3157:
3153:
3146:
3140:
3137:
3134:
3128:
3122:
3117:
3113:
3092:
3089:
3086:
3083:
3077:
3073:
3066:
3062:
3058:
3055:
3052:
3044:
3039:
3034:
3030:
3025:
3021:
3015:
2983:
2970:
2965:
2952:
2949:
2946:
2940:
2936:
2932:
2927:
2922:
2917:
2913:
2908:
2904:
2898:
2892:
2878:
2871:
2861:
2848:
2845:
2842:
2836:
2832:
2826:
2820:
2816:
2810:
2806:
2801:
2797:
2788:
2769:
2763:
2759:
2748:
2742:
2738:
2734:, the number
2731:
2712:
2709:
2704:
2700:
2694:
2688:
2684:
2678:
2663:
2646:
2640:
2633:
2629:
2625:
2622:
2617:
2613:
2606:
2604:
2599:
2592:
2586:
2581:
2576:
2572:
2568:
2563:
2559:
2552:
2550:
2545:
2533:
2530:
2515:
2511:
2507:
2504:
2499:
2495:
2491:
2488:
2483:
2479:
2470:
2464:
2455:
2449:
2446:
2433:
2430:
2427:
2424:
2421:
2417:
2411:
2407:
2402:
2396:
2393:
2390:
2386:
2381:
2378:
2370:
2367:
2347:
2344:
2342:
2337:
2334:
2331:
2328:
2325:
2318:
2315:
2313:
2308:
2305:
2302:
2295:
2286:
2274:
2270:
2260:
2250:
2236:
2219:
2214:
2211:
2208:
2204:
2200:
2195:
2192:
2189:
2185:
2181:
2179:
2169:
2166:
2163:
2159:
2155:
2152:
2147:
2144:
2141:
2137:
2133:
2130:
2125:
2122:
2119:
2115:
2111:
2108:
2103:
2100:
2097:
2093:
2089:
2086:
2084:
2071:
2068:
2065:
2061:
2057:
2052:
2049:
2046:
2042:
2035:
2032:
2024:
2021:
2018:
2014:
2010:
2005:
2002:
1999:
1995:
1988:
1985:
1983:
1973:
1969:
1965:
1962:
1957:
1953:
1949:
1946:
1944:
1937:
1933:
1920:
1917:
1902:
1898:
1894:
1891:
1886:
1882:
1878:
1875:
1870:
1866:
1857:
1846:
1829:
1824:
1821:
1818:
1814:
1810:
1805:
1802:
1799:
1795:
1791:
1789:
1782:
1778:
1770:
1765:
1762:
1759:
1755:
1751:
1746:
1743:
1740:
1736:
1732:
1730:
1723:
1719:
1706:
1684:
1679:
1676:
1673:
1669:
1665:
1660:
1657:
1654:
1650:
1646:
1641:
1637:
1616:
1613:
1610:
1607:
1602:
1598:
1580:
1567:
1561:
1558:
1555:
1552:
1545:
1542:
1534:
1531:
1525:
1520:
1516:
1509:
1503:
1496:
1493:
1485:
1482:
1476:
1471:
1467:
1460:
1455:
1451:
1442:
1426:
1423:
1419:
1415:
1412:
1409:
1400:
1387:
1384:
1381:
1377:
1374:
1371:
1366:
1363:
1358:
1355:
1352:
1349:
1346:
1343:
1338:
1332:
1327:
1324:
1318:
1315:
1307:
1305:
1297:
1292:
1279:
1276:
1272:
1269:
1264:
1258:
1253:
1250:
1244:
1241:
1233:
1230:
1217:
1211:
1204:
1200:
1196:
1191:
1187:
1180:
1174:
1171:
1168:
1161:
1157:
1153:
1148:
1144:
1137:
1132:
1128:
1119:
1117:
1113:
1109:
1105:
1101:
1097:
1093:
1082:
1070:
1063:
1059:
1054:
1050:
1048:
1047:Ădouard Lucas
1043:
1031:
1024:
1011:
1008:
1005:
1002:
1001:
1000:
998:
993:
990:unrealistic)
989:
985:
981:
976:
975:
963:
958:
957:
951:
946:
937:
935:
930:
928:
922:
920:
916:
915:
914:Natya Shastra
910:
903:
899:
892:
888:
880:
876:
867:
859:
854:
849:
845:
836:
832:
820:
811:
802:
796:
791:
769:
766:
763:
760:
757:
754:
751:
748:
745:
742:
739:
736:
733:
730:
727:
724:
721:
718:
715:
712:
711:
704:
700:
694:
690:
684:
680:
674:
670:
664:
660:
654:
650:
644:
640:
634:
630:
624:
620:
614:
610:
604:
600:
594:
590:
584:
580:
574:
570:
564:
560:
554:
550:
544:
540:
534:
530:
524:
520:
514:
510:
509:
506:
505:
504:
501:
492:
488:
484:is valid for
469:
466:
463:
459:
455:
450:
447:
444:
440:
436:
431:
427:
406:
403:
400:
395:
391:
387:
382:
378:
357:
354:
349:
345:
335:
331:
312:
309:
306:
302:
298:
293:
290:
287:
283:
279:
274:
270:
249:
246:
243:
238:
234:
229:
226:
223:
218:
214:
206:
198:
197:circular arcs
194:
193:golden spiral
189:
180:
178:
174:
170:
169:Lucas numbers
154:
150:
145:
143:
139:
135:
131:
127:
123:
119:
115:
112:
108:
104:
103:
97:
94:
93:
87:
83:
79:
75:
67:
62:
55:
54:
53:
46:
37:
33:
29:
22:
28876:
28835:
28816:
28808:
28657:Power series
28399:Lucas number
28378:
28351:Powers of 10
28331:Cubic number
28164:
28145:Pisot number
27825:Transposable
27689:Narcissistic
27596:Digital root
27516:Super-Poulet
27476:JordanâPĂłlya
27425:prime factor
27330:Noncototient
27297:Almost prime
27279:Superperfect
27254:Refactorable
27249:Quasiperfect
27224:Hyperperfect
27065:Pseudoprimes
27036:WallâSunâSun
26971:Ordered Bell
26941:FussâCatalan
26853:non-centered
26803:Dodecahedral
26780:non-centered
26666:non-centered
26568:Wolstenholme
26431:
26313:× 2 ± 1
26310:
26309:Of the form
26276:Eighth power
26256:Fourth power
26147:
26132:
26119:at MathPages
26081:
26070:
26046:
26042:Livio, Mario
26023:
26004:
25977:
25974:BĂłna, MiklĂłs
25955:
25932:
25910:
25898:
25882:11250/180781
25864:
25860:
25854:
25820:
25814:
25805:
25798:
25786:
25776:, retrieved
25769:the original
25764:
25739:
25730:
25726:
25714:
25691:
25673:
25661:
25645:
25641:
25635:
25613:
25607:
25598:
25592:, retrieved
25577:
25570:
25558:
25546:
25529:
25522:
25513:
25504:
25493:
25487:
25469:
25463:
25452:the original
25439:
25435:
25422:
25412:, retrieved
25406:
25399:
25390:
25383:
25374:
25369:
25360:
25354:
25346:
25337:
25331:
25325:
25307:
25298:
25266:
25259:
25246:(4): 39â40,
25243:
25239:
25233:
25224:
25221:Modern Logic
25220:
25214:
25206:
25188:
25182:
25177:, p. 7.
25170:
25154:
25150:
25132:
25120:
25107:
25083:
25079:
25073:
25063:
25055:
25051:
25043:
25037:
25027:
25020:
25016:
25008:
25002:
24990:
24978:
24966:
24948:
24939:
24927:
24914:
24908:, UK: Surrey
24904:
24902:Knott, Ron,
24897:
24873:
24869:
24859:
24843:
24839:
24829:
24789:
24785:
24779:
24770:
24766:
24753:
24721:math/0403046
24711:
24707:
24701:
24692:
24688:
24682:
24658:
24654:
24648:
24630:
24624:
24614:, retrieved
24607:the original
24584:
24564:
24552:
24539:
24531:
24527:
24490:
24484:
24437:
24427:, retrieved
24423:the original
24417:
24410:
24401:
24392:
24374:
24364:
24348:
24342:
24336:
24324:
24307:
24291:
24259:
24255:
24249:
24238:
24200:
24193:
24175:
24169:
24164:, p. 4.
24157:
24145:
24135:
24124:. Retrieved
24120:
24111:
24100:, retrieved
24095:
24089:
24076:
24067:
24062:
24044:
24038:
24026:
24013:
24001:
23988:
23976:
23964:
23938:
23932:
23909:
23902:
23893:
23879:
23870:
23856:
23854:Knott, Ron,
23849:
23838:, retrieved
23832:
23822:
23804:
23798:
23787:, retrieved
23780:
23771:
23759:
23748:, retrieved
23741:
23728:
23716:
23707:
23701:
23679:
23673:
23667:
23658:
23654:
23651:
23645:
23623:ed) ...
23621:
23606:
23596:
23584:
23572:
23567:, p. 3.
23560:
23548:
23536:
23533:
23518:
23487:
23483:
23455:
23431:
23391:
23264:
23253:
23043:Y chromosome
23039:X chromosome
23036:
23019:
23015:
23007:
23003:
22996:
22992:
22988:
22965:
22950:
22946:
22942:
22931:
22927:
22923:
22919:
22915:golden angle
22896:
22825:
22814:Helmut Vogel
22803:
22796:
22775:
22767:parastichies
22747:golden ratio
22720:
22601:
22346:
22302:
22294:
22290:
22262:
22258:
22251:
22232:
22224:
22220:
22202:
22188:
22184:
22180:
22168:
22155:
22151:
22133:
22125:
22121:
22095:
22087:
22018:
22011:
21973:
21969:
21965:compositions
21958:
21955:compositions
21939:
21935:
21837:
21771:
21406:
21386:Applications
21378:
21369:
21365:
21361:
21357:
21353:
21349:
21322:
21318:
21312:
21308:
21301:
21297:Pell numbers
21282:
21278:
21272:
21268:
21261:
21251:
21241:
21230:real numbers
21216:
21201:
21178:finite field
21166:open problem
21152:
21140:
21129:
21119:
21112:
21108:
21105:
20878:
20871:
20867:
20860:
20856:
20849:
20479:
20472:
20468:
20467:
20091:
20084:
20080:
20079:
19709:
19702:
19698:
19697:
19315:
19308:
19304:
19303:
19010:
19007:
19000:
18919:
18912:
18440:
18307:
18093:
18084:
18075:
18062:
18053:
18044:
18037:
18034:
18018:
18014:Vern Hoggatt
18003:
17992:
17982:
17979:
17961:
17954:
17950:
17943:
17939:
17937:
17934:
17920:
17918:
17893:
17733:
17726:
17721:
17717:
17704:
17615:
17541:
17351:
17343:
17333:
17325:
17321:
17305:
17301:
17291: 5. If
17279:prime number
17276:
17269:
17115:. That is,
17044:
16915:
16876:
16650:
16649:
16638:
16474:
16118:
15976:golden ratio
15971:
15969:
15877:
15663:
15514:
15391:
15328:
15236:
15015:is given by
15010:
14954:
14384:
14317:
14244:
14088:
14085:power series
14078:
14062:
14059:
13907:
13904:
13752:
13749:
13276:
13105:Lucas number
13094:
13090:
12812:
12588:
12574:
11921:
11855:
11492:
11386:
11163:
11156:
11033:
10871:
10859:
10731:
10614:
10607:
10574:index up to
10565:
10526:index up to
10363:
10254:
10061:
9845:
9766:
9758:
9287:
8552:
8527:
8523:
8516:
8205:
8201:
8198:
7987:
7969:
7960:
7956:
7953:
7950:
7848:
7837:
7833:
7827:For a given
7826:
7683:
7676:
7672:
7666:
7662:
7655:
7643:
7639:
7633:
7629:
7624:
7620:
7609:
7480:
7469:
7455:
7452:
7355:
6831:
6824:
6688:
6685:
6577:
6567:
6560:
6345:
5879:
5759:
5644:eigenvectors
5487:
5407:
5188:
5181:
5173:
4920:
4806:
4557:
4473:
4467:
4464:
4053:
3976:coefficients
3922:
3878:
3763:In general,
3762:
3650:
3547:
3465:
3461:
3451:
3440:
3387:
3383:
3332:
3330:
2966:
2876:
2869:
2862:
2740:
2736:
2729:
2669:
2534:
2531:
2471:
2462:
2453:
2450:
2447:
2371:
2368:
2287:
2272:
2268:
2258:
2248:
2237:
1921:
1918:
1858:
1847:
1707:
1581:
1443:
1401:
1308:
1296:golden ratio
1293:
1234:
1231:
1120:
1103:
1089:
1081:Golden ratio
1065:
1061:
1044:
1029:
1022:
1015:
997:math problem
988:biologically
979:
969:
932:
927:mÄtrÄ-váčttas
926:
924:
912:
909:Bharata Muni
901:
897:
890:
886:
878:
874:
865:
855:
847:
843:
828:
818:
809:
800:
702:
692:
682:
672:
662:
652:
642:
632:
622:
612:
602:
592:
582:
572:
562:
552:
542:
532:
522:
512:
496:
493:
486:
336:
329:
202:
149:golden ratio
146:
100:
98:
71:
41:
35:
27:
25:
28810:Liber Abaci
28524:Conditional
28512:Convergence
28503:Telescoping
28488:Alternating
28404:Pell number
28214:Pell number
27846:Extravagant
27841:Equidigital
27796:permutation
27755:Palindromic
27729:Automorphic
27627:Sum-product
27606:Sum-product
27561:Persistence
27456:ErdĆsâWoods
27378:Untouchable
27259:Semiperfect
27209:Hemiperfect
26870:Tesseractic
26808:Icosahedral
26788:Tetrahedral
26719:Dodecagonal
26420:Recursively
26291:Prime power
26266:Sixth power
26261:Fifth power
26241:Power of 10
26199:Classes of
26133:In Our Time
25925:Works cited
25498:(7): 525â32
25363:(3): 265â69
25227:(3): 345â55
25157:: 169â177,
25062:10000 <
25047:, Red golpe
25012:, Mersennus
24773:(2): 98â108
24661:: 537â540,
24098:(4): 417â19
23840:27 November
23789:28 November
23764:Sigler 2002
23553:Sigler 2002
22970:(which are
22817: [
22799:= 1 ... 500
22781:free groups
22771:phyllotaxis
22611:binary tree
22347:one or more
22330:his solving
22091:recursively
22040:= 1+1+1+1+1
21786:= 1+1+1+1+1
21391:Mathematics
20482:⥠1 (mod 4)
20094:⥠1 (mod 4)
19712:⥠3 (mod 4)
19318:⥠3 (mod 4)
18010:conjectured
14268:satisfying
14069:Matrix form
8743:cardinality
8538:memoization
8519:recursively
7612:convergents
7461:determinant
7453:The matrix
5884:th term is
5480:eigenvalues
5178:Matrix form
1090:Like every
982:, 1202) by
974:Liber Abaci
956:Liber Abaci
934:Hemachandra
92:Liber Abaci
28896:Categories
28549:Convergent
28493:Convergent
28058:Graphemics
27931:Pernicious
27785:Undulating
27760:Pandigital
27734:Trimorphic
27335:Nontotient
27184:Arithmetic
26798:Octahedral
26699:Heptagonal
26689:Pentagonal
26674:Triangular
26515:SierpiĆski
26437:Jacobsthal
26236:Power of 3
26231:Power of 2
25915:Livio 2003
25903:Livio 2003
25791:Livio 2003
25778:2016-09-03
25666:Livio 2003
25594:2022-10-30
25563:Livio 2003
25551:Livio 2003
25414:2022-08-15
25175:Lucas 1891
25066:< 50000
25030:< 10000
24876:: 107â24,
24708:Ann. Math.
24633:(9): 133,
24616:2022-11-23
24429:2007-02-23
24162:Lucas 1891
24126:2024-03-22
23721:Livio 2003
23638:Livio 2003
23565:Lucas 1891
23378:References
23321:Mario Merz
23284:conversion
23282:Since the
22751:pentagonal
22749:-related)
22704:See also:
22589:merge sort
22538:ring lemma
22362:hypotenuse
22192:for every
22177:for which
21933:twos from
21122:< 50000
20469:Example 4.
20081:Example 3.
19699:Example 2.
19305:Example 1.
18917:such that
18006:triangular
17270:for every
17042:function.
16641:irrational
15517:reciprocal
11003:partitions
7975:identities
7671:= 1 + 1 /
3340:asymptotic
948:A page of
866:misrau cha
798:Thirteen (
788:See also:
183:Definition
28795:Fibonacci
28580:Divergent
28498:Divergent
28360:Advanced
28336:Factorial
28284:Sequences
28175:Rectangle
27815:Parasitic
27664:Factorion
27591:Digit sum
27583:Digit sum
27401:Fortunate
27388:Primorial
27302:Semiprime
27239:Practical
27204:Descartes
27199:Deficient
27189:Betrothed
27031:Wieferich
26860:Pentatope
26823:pyramidal
26714:Decagonal
26709:Nonagonal
26704:Octagonal
26694:Hexagonal
26553:Practical
26500:Congruent
26432:Fibonacci
26396:Loeschian
26154:EMS Press
26044:(2003) ,
25340:: 263â266
25291:pp. 73â74
24882:1787-6117
24822:121789033
24806:1973-4409
24461:ε
24458:−
24284:116536130
24240:MathWorld
24102:April 11,
23981:Ball 2003
23969:Ball 2003
23750:4 January
23589:BĂłna 2011
23407:Citations
23315:economics
23291:2 number
23258:, is the
22980:drone bee
22850:φ
22845:π
22833:θ
22810:sunflower
22739:honeybees
22735:pine cone
22731:artichoke
22727:pineapple
22514:triangle.
22250:{1, ...,
22198:bijection
22167:{1, ...,
22069:= 1+1+1+2
22055:= 1+1+2+1
22050:= 1+2+1+1
22045:= 2+1+1+1
21994:×
21838:which is
21810:= 1+1+1+2
21805:= 1+1+2+1
21800:= 1+2+1+1
21795:= 2+1+1+1
21708:∞
21693:∑
21686:⋯
21639:⋯
21551:−
21545:−
21498:−
21492:−
21462:−
21437:∑
21089:⋅
21083:⋅
21001:⋅
20817:≡
20758:≡
20577:−
20435:≡
20376:−
20373:≡
20203:−
20189:−
20133:−
20047:≡
19988:≡
19874:−
19665:−
19662:≡
19603:−
19600:≡
19497:−
19483:−
19427:−
19357:−
19261:≡
19224:∓
19152:≡
19115:±
19058:≡
19038:±
19008:Also, if
18958:≡
18935:−
18746:−
18582:−
18486:−
18405:≡
18382:−
18330:≡
18265:±
18262:≡
18246:−
18220:±
18217:≡
17817:≡
17801:−
17697:composite
17643:−
17631:∣
17573:−
17561:∣
17544:piecewise
17502:∣
17496:⇒
17472:±
17469:≡
17451:−
17440:∣
17434:⇒
17410:±
17407:≡
17384:∣
17378:⇒
17311:, and if
17016:…
16978:…
16826:−
16806:−
16756:∑
16715:−
16677:∞
16662:∑
16625:…
16594:∞
16579:∑
16565:−
16544:∞
16529:∑
16503:∞
16488:∑
16439:ψ
16435:−
16419:ψ
16413:−
16398:ψ
16394:−
16378:ψ
16302:−
16281:∞
16266:∑
16238:with the
16212:ψ
16202:−
16190:ψ
16143:∞
16128:∑
16093:−
16038:∑
16013:−
16002:∞
15987:∑
15930:−
15902:∞
15887:∑
15826:−
15798:ϑ
15794:−
15767:−
15739:ϑ
15688:∞
15673:∑
15623:−
15595:ϑ
15568:−
15547:∞
15532:∑
15493:…
15360:−
15262:−
15251:−
15248:↦
15232:conjugate
15206:−
15180:ψ
15122:φ
15091:ψ
15088:−
15076:−
15067:φ
15064:−
14993:≥
14903:−
14892:−
14884:−
14873:−
14855:∞
14840:∑
14820:−
14763:−
14750:∞
14735:∑
14731:−
14713:−
14700:∞
14685:∑
14681:−
14656:∞
14641:∑
14599:∞
14584:∑
14580:−
14549:∞
14534:∑
14530:−
14505:∞
14490:∑
14454:−
14448:−
14405:−
14399:−
14355:−
14349:−
14300:φ
14228:…
14127:∞
14112:∑
14038:−
14025:−
13938:∑
13883:−
13833:−
13783:∑
13745:factorize
13641:−
13372:−
13239:−
13197:−
13043:−
12983:−
12970:−
12905:−
12881:−
12846:−
12770:−
12756:−
12742:−
12715:−
12671:−
12657:−
12643:−
12616:−
12545:ψ
12542:−
12539:φ
12516:−
12510:ψ
12507:φ
12477:−
12435:ψ
12432:−
12429:φ
12423:−
12408:ψ
12404:−
12389:φ
12372:ψ
12369:φ
12364:ψ
12361:−
12346:ψ
12339:φ
12321:φ
12317:−
12298:φ
12295:−
12287:−
12272:ψ
12265:−
12259:ψ
12256:−
12248:−
12233:φ
12210:−
12207:ψ
12199:−
12184:ψ
12177:−
12168:−
12165:φ
12157:−
12142:φ
12114:ψ
12107:⋯
12095:ψ
12088:ψ
12074:−
12065:φ
12058:⋯
12046:φ
12039:φ
11980:−
11931:∑
11898:ψ
11894:−
11885:φ
11764:∑
11626:∑
11544:∑
11476:−
11421:∑
11369:−
11282:∑
11222:−
11173:∑
11122:−
11116:−
11096:, is the
11062:∞
11047:∑
11016:−
11001:(ordered
10939:
10901:∈
10811:−
10754:×
10624:∑
10545:−
10507:−
10452:∑
10391:−
10373:∑
10346:−
10297:∑
10113:−
9829:−
9780:∑
9712:−
9693:−
9660:−
9627:−
9596:−
9570:−
9336:−
9317:−
8595:−
8473:−
8407:−
8343:−
8314:−
8279:−
8232:−
8140:−
8088:−
8058:−
8040:−
7915:−
7907:−
7864:−
7797:−
7578:⋱
7488:φ
7412:−
7404:φ
7401:−
7395:−
7386:φ
7305:φ
7297:−
7285:−
7281:φ
7239:−
7231:φ
7228:−
7202:φ
7165:−
7161:φ
7157:−
7152:φ
7090:−
7069:−
7061:φ
7058:−
7032:φ
6966:−
6952:Λ
6783:−
6779:φ
6775:−
6770:φ
6735:−
6731:φ
6727:−
6710:φ
6696:Λ
6660:−
6646:Λ
6610:−
6602:Λ
6509:−
6459:−
6308:−
6304:φ
6300:−
6253:−
6203:−
6188:φ
6080:→
6077:ν
6066:−
6058:φ
6055:−
6037:−
6031:→
6028:μ
6016:φ
5987:→
5984:ν
5956:−
5950:→
5947:μ
5903:→
5860:→
5857:ν
5839:−
5833:→
5830:μ
5775:→
5726:−
5722:φ
5718:−
5700:→
5697:ν
5676:φ
5659:→
5656:μ
5612:−
5579:−
5575:φ
5571:−
5565:ψ
5499:φ
5457:→
5423:→
5382:→
5349:→
5140:−
5094:−
5024:−
4966:−
4952:φ
4930:φ
4883:−
4847:±
4815:φ
4782:φ
4751:φ
4703:−
4699:φ
4682:−
4676:−
4667:φ
4623:−
4520:−
4443:−
4429:ψ
4407:ψ
4380:ψ
4368:ψ
4347:φ
4336:−
4330:ψ
4294:φ
4259:φ
4248:−
4218:φ
4210:−
4187:φ
4168:φ
4160:−
4140:φ
4123:φ
4112:−
4098:φ
4067:φ
4049:induction
4023:−
4009:φ
3987:φ
3958:φ
3932:φ
3900:φ
3888:φ
3847:φ
3821:φ
3782:∞
3779:→
3744:φ
3726:−
3705:, unless
3634:φ
3596:∞
3593:→
3566::
3563:φ
3553:converges
3520:
3512:φ
3509:
3494:φ
3491:
3419:≈
3416:φ
3413:
3351:φ
3327:Magnitude
3311:…
3299:φ
3293:
3263:…
3251:φ
3245:
3219:φ
3213:
3186:
3167:φ
3161:
3141:
3123:
3118:φ
3087:≥
3040:
3035:φ
2950:≥
2923:
2918:φ
2846:≥
2817:φ
2760:φ
2685:ψ
2626:−
2623:φ
2582:ψ
2569:−
2512:ψ
2496:φ
2428:−
2397:ψ
2394:−
2391:φ
2335:ψ
2326:φ
2212:−
2193:−
2167:−
2160:ψ
2145:−
2138:φ
2123:−
2116:ψ
2101:−
2094:φ
2069:−
2062:ψ
2050:−
2043:ψ
2022:−
2015:φ
2003:−
1996:φ
1970:ψ
1954:φ
1899:ψ
1883:φ
1822:−
1815:ψ
1803:−
1796:ψ
1779:ψ
1763:−
1756:φ
1744:−
1737:φ
1720:φ
1677:−
1658:−
1629:and thus
1559:−
1556:φ
1543:−
1535:φ
1532:−
1526:−
1517:φ
1494:−
1486:φ
1483:−
1477:−
1468:φ
1424:−
1420:φ
1416:−
1410:ψ
1385:…
1375:−
1372:≈
1367:φ
1359:−
1353:φ
1350:−
1328:−
1316:ψ
1304:conjugate
1280:…
1270:≈
1242:φ
1201:ψ
1197:−
1188:φ
1175:ψ
1172:−
1169:φ
1158:ψ
1154:−
1145:φ
984:Fibonacci
960:from the
950:Fibonacci
919:Virahanka
841:units is
467:−
448:−
310:−
291:−
142:pine cone
138:artichoke
134:pineapple
86:Fibonacci
28829:Theories
28753:Category
28519:Absolute
28195:Triangle
27887:Friedman
27820:Primeval
27765:Repdigit
27722:-related
27669:Kaprekar
27643:Meertens
27566:Additive
27553:dynamics
27461:Friendly
27373:Sociable
27363:Amicable
27174:Abundant
27154:dynamics
26976:Schröder
26966:Narayana
26936:Eulerian
26926:Delannoy
26921:Dedekind
26742:centered
26608:centered
26495:Amenable
26452:Narayana
26442:Leonardo
26338:Mersenne
26286:Powerful
26226:Achilles
25976:(2011),
25847:35413237
25685:(1990),
25306:(1997),
25252:30215477
24947:(1996),
24840:Integers
24746:10266596
24572:(2018),
24400:(2000),
24143:(1978),
23878:(1996),
23604:(1968),
23516:(2006),
23342:See also
23293:register
22949:(
22930:(
22922:(
22769:(spiral
22662:compands
22619:AVL tree
21474:⌋
21453:⌊
21225:numbers.
21147:periodic
21117:for all
20850:For odd
19254:if
19145:if
18255:if
18210:if
18187:if
17973:runs of
17707:. When
17342:divides
17319:divides
17299:divides
15469:0.998999
14060:Putting
12019:becomes
11617:to give
10728:-th and
7964:for any
7846:method.
4800:via the
3308:0.208987
3260:0.481211
3074:⌋
3026:⌊
2937:⌉
2909:⌊
2833:⌉
2811:⌊
2787:rounding
2727:for all
1092:sequence
109:and the
82:Sanskrit
32:sequence
28850:Related
28529:Uniform
28180:Rhombus
28060:related
28024:related
27988:related
27986:Sorting
27871:Vampire
27856:Harshad
27798:related
27770:Repunit
27684:Lychrel
27659:Dudeney
27511:StĂžrmer
27506:Sphenic
27491:Regular
27429:divisor
27368:Perfect
27264:Sublime
27234:Perfect
26961:Motzkin
26916:Catalan
26457:Padovan
26391:Leyland
26386:Idoneal
26381:Hilbert
26353:Woodall
26156:, 2001
26136:at the
26111:YouTube
25891:8550726
25827:Bibcode
25697:101â107
25388:"IFF",
25286:2131318
25163:2278830
25115:(ed.),
25100:2325076
24922:(ed.),
24890:2753031
24852:2988067
24814:1765401
24726:Bibcode
24695:: 81â96
24675:0163867
24603:3821829
24547:(ed.),
24519:0668957
24357:0999451
24315:(ed.),
24300:Borwein
24276:3618079
24021:(ed.),
23996:(ed.),
23686:: 232,
23426:(ed.),
22806:florets
22755:daisies
22649:in the
22187:+ 1} â
22105:subsets
22101:strings
22079:= 1+2+2
22074:= 2+1+2
22060:= 2+2+1
21983:of the
21944:terms.
21829:= 1+2+2
21824:= 2+1+2
21819:= 2+2+1
21364:â 2) +
20814:1860500
17047:coprime
17038:is the
16475:So the
15230:is its
13099:is the
11237:Adding
10604:is the
10562:is the
8741:is the
7648:is the
6686:where
5482:of the
5478:. The
2747:integer
1378:0.61803
1302:is its
1294:is the
1273:1.61803
1058:problem
907:cases.
872:beats (
858:Pingala
779:History
120:called
78:Pingala
66:squares
49:
28821:(1225)
28813:(1202)
28481:Series
28288:series
28234:etc...
28231:Nickel
28228:Copper
28225:Bronze
28209:Silver
28190:Spiral
27926:Odious
27851:Frugal
27805:Cyclic
27794:Digit-
27501:Smooth
27486:Pronic
27446:Cyclic
27423:Other
27396:Euclid
27046:Wilson
27020:Primes
26679:Square
26548:Polite
26510:Riesel
26505:Knödel
26467:Perrin
26348:Thabit
26333:Fermat
26323:Cullen
26246:Square
26214:Powers
26088:
26058:
26030:
26012:
25984:
25962:
25943:
25889:
25845:
25703:
25624:
25585:
25476:
25314:
25284:
25274:
25250:
25195:
25161:
25098:
24955:
24888:
24880:
24850:
24846:: A7,
24820:
24812:
24804:
24744:
24673:
24637:
24601:
24591:
24517:
24381:
24355:
24296:Landau
24282:
24274:
24208:
24182:
24070:, 1611
24051:
23953:
23917:
23886:
23811:
23614:
23526:
23463:
23310:ratio.
23256:> 1
23241:optics
22897:where
22867:
22743:Kepler
22694:Nature
22615:height
22521:is an
22098:binary
21259:, and
21054:
21032:
21010:
20972:
20950:
20928:
20906:
20755:710645
17995:square
17289:modulo
17277:Every
17034:where
16331:since
15972:nested
15482:998999
15392:Using
15114:where
13088:where
7459:has a
5622:
5537:
5484:matrix
4470:< 1
4045:proved
3443:> 1
3422:0.2090
3331:Since
3275:, and
3105:where
3081:
2944:
2840:
2670:Since
2532:where
1402:Since
1298:, and
1232:where
992:rabbit
940:Europe
489:> 2
332:> 1
118:graphs
116:, and
28802:Books
28424:array
28304:Basic
28155:Angle
27967:Prime
27962:Lucky
27951:sieve
27880:Other
27866:Smith
27746:Digit
27704:Happy
27679:Keith
27652:Other
27496:Rough
27466:Giuga
26931:Euler
26793:Cubic
26447:Lucas
26343:Proth
25887:S2CID
25843:S2CID
25772:(PDF)
25761:(PDF)
25723:(PDF)
25534:(PDF)
25514:Maths
25455:(PDF)
25432:(PDF)
25407:Story
25248:JSTOR
25096:JSTOR
25060:with
25025:with
24818:S2CID
24763:(PDF)
24742:S2CID
24716:arXiv
24610:(PDF)
24579:, in
24577:(PDF)
24280:S2CID
24272:JSTOR
24150:(PDF)
24086:(PDF)
23895:abaci
23738:(PDF)
23682:(3),
23397:morae
23289:radix
23234:Other
22911:angle
22821:]
22666:Ό-law
22658:Amiga
22364:of a
22314:s or
22287:s is
22283:s or
21347:have
21299:have
21239:have
21172:of a
17925:prime
17725:(mod
17703:is a
15466:0.001
15455:0.001
11005:) of
10855:areas
8745:of a
8526:(log
7985:into
7836:(log
4638:is a
1382:39887
1277:39887
784:India
770:4181
767:2584
764:1597
503:are:
30:is a
28364:list
28286:and
28160:Base
28150:Gold
27921:Evil
27601:Self
27551:and
27441:Blum
27152:and
26956:Lobb
26911:Cake
26906:Bell
26656:Star
26563:Ulam
26462:Pell
26251:Cube
26161:OEIS
26086:ISBN
26056:ISBN
26028:ISBN
26010:ISBN
25982:ISBN
25960:ISBN
25941:ISBN
25701:ISBN
25622:ISBN
25583:ISBN
25474:ISBN
25312:ISBN
25272:ISBN
25193:ISBN
25121:The
24953:ISBN
24928:The
24878:ISSN
24802:ISSN
24635:ISBN
24589:ISBN
24553:The
24379:ISBN
24325:The
24206:ISBN
24180:ISBN
24104:2012
24049:ISBN
24027:The
24002:The
23951:ISBN
23915:ISBN
23884:ISBN
23842:2018
23809:ISBN
23791:2018
23752:2019
23746:: 41
23612:ISBN
23524:ISBN
23461:ISBN
23432:The
22968:bees
22934:+ 1)
22761:and
22654:8SVX
22544:and
22529:for
22517:The
22196:. A
21372:â 3)
21356:) =
21343:and
20711:610.
20475:= 29
20087:= 13
19705:= 11
18082:and
18060:and
17082:and
15479:1000
15011:The
14982:for
14289:<
14079:The
13103:-th
12531:and
10734:+ 1)
10610:+ 1)
10572:even
10443:and
9769:+ 2)
7610:The
5557:and
5491:are
4322:For
3678:and
3454:base
2705:<
2460:and
2281:and
2256:and
2242:and
1852:and
1701:and
1586:and
1114:and
761:987
758:610
755:377
752:233
749:144
327:for
262:and
28039:Ban
27427:or
26946:Lah
26138:BBC
26130:on
26109:on
25877:hdl
25869:doi
25865:154
25835:doi
25687:"4"
25650:doi
25444:doi
25440:178
25338:146
25088:doi
24844:11a
24794:doi
24734:doi
24663:doi
24505:hdl
24495:doi
24483:",
24349:308
24264:doi
23943:doi
23688:doi
23539:= 7
23492:doi
23295:in
23267:= 1
23239:In
22651:IFF
22576:of
22308:= 6
22248:of
22238:= 5
22165:of
22139:= 8
22024:= 8
22015:5+1
21307:= 2
21257:= 3
21247:= 1
21092:61.
21072:610
21048:233
20828:mod
20769:mod
20690:377
20446:mod
20432:845
20387:mod
20370:320
20326:13.
20058:mod
20044:320
19999:mod
19985:125
19676:mod
19614:mod
19311:= 7
19272:mod
19240:mod
19163:mod
19131:mod
19013:â 5
18969:mod
18895:89.
18416:mod
18369:and
18356:mod
18308:If
18276:mod
18231:mod
18012:by
17988:= 8
17967:= 3
17868:mod
17713:bit
17695:is
17483:mod
17421:mod
17336:= 5
17208:gcd
17167:gcd
17126:gcd
17036:gcd
16992:gcd
16933:gcd
16643:by
16479:is
14065:= 2
13743:to
10936:Seq
10524:odd
9644:or
9585:or
8747:set
8540:).
8521:in
7992:),
7990:+ 1
6565:of
4773:in
4602:or
4479:is
4056:â„ 1
4051:on
4047:by
3772:lim
3586:lim
3517:log
3506:log
3482:log
3404:log
3342:to
3338:is
3284:log
3204:log
3177:log
3114:log
3031:log
2914:log
2879:â„ 8
2872:â„ 4
2749:to
2732:â„ 0
2264:= 1
2254:= 0
2238:If
1032:â 1
1025:â 2
952:'s
746:89
743:55
740:34
737:21
734:13
491:.
28898::
26152:,
26146:,
26054:,
25998:;
25939:,
25885:,
25875:,
25863:,
25841:,
25833:,
25763:,
25748:^
25729:,
25725:,
25699:,
25689:,
25681:;
25646:44
25644:,
25620:,
25616:,
25597:,
25536:,
25512:,
25438:,
25434:,
25282:MR
25280:,
25244:30
25242:,
25223:,
25219:,
25159:MR
25155:71
25153:,
25149:,
25119:,
25094:,
25084:99
25082:,
24926:,
24886:MR
24884:,
24874:37
24872:,
24868:,
24848:MR
24842:,
24838:,
24816:,
24810:MR
24808:,
24800:,
24790:49
24788:,
24771:27
24769:,
24765:,
24740:,
24732:,
24724:,
24710:,
24693:17
24691:,
24671:MR
24669:,
24659:39
24657:,
24599:MR
24597:,
24551:,
24515:MR
24513:,
24503:,
24491:25
24489:,
24373:,
24353:MR
24347:,
24323:,
24278:,
24270:,
24260:79
24258:,
24237:,
24233:,
24219:^
24119:.
24096:10
24094:,
24088:,
24025:,
24000:,
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6827:n
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6800:1
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6650:n
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6639:=
6630:n
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6613:1
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6596:=
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6568:A
6547:.
6542:n
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6069:n
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6052:(
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6020:n
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6000:=
5976:n
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5965:5
5961:1
5939:n
5935:A
5928:5
5924:1
5919:=
5910:n
5900:F
5882:n
5866:,
5848:5
5844:1
5821:5
5817:1
5812:=
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5801:0
5798:1
5793:(
5787:=
5782:0
5772:F
5746:.
5740:)
5735:1
5729:1
5712:(
5706:=
5690:,
5684:)
5679:1
5671:(
5665:=
5628:)
5617:5
5609:1
5604:(
5596:2
5593:1
5587:=
5582:1
5568:=
5543:)
5532:5
5527:+
5524:1
5519:(
5511:2
5508:1
5502:=
5488:A
5464:0
5454:F
5445:n
5440:A
5435:=
5430:n
5420:F
5394:,
5389:k
5379:F
5371:A
5367:=
5362:1
5359:+
5356:k
5346:F
5319:)
5312:k
5308:F
5302:1
5299:+
5296:k
5292:F
5286:(
5278:)
5272:0
5267:1
5260:1
5255:1
5249:(
5244:=
5238:)
5231:1
5228:+
5225:k
5221:F
5215:2
5212:+
5209:k
5205:F
5199:(
5159:.
5153:2
5149:)
5143:1
5137:n
5133:F
5129:2
5126:+
5121:n
5117:F
5113:(
5110:=
5105:n
5101:)
5097:1
5091:(
5088:4
5085:+
5080:2
5073:n
5069:F
5063:5
5040:2
5036:/
5032:)
5027:1
5021:n
5017:F
5013:2
5010:+
5005:n
5001:F
4997:+
4992:5
4985:n
4981:F
4977:(
4974:=
4969:1
4963:n
4959:F
4955:+
4947:n
4943:F
4939:=
4934:n
4907:.
4902:2
4894:n
4890:)
4886:1
4880:(
4877:4
4874:+
4869:2
4862:n
4858:F
4852:5
4842:5
4835:n
4831:F
4824:=
4819:n
4786:n
4755:n
4745:5
4721:5
4715:/
4711:)
4706:n
4693:n
4689:)
4685:1
4679:(
4671:n
4663:(
4660:=
4655:n
4651:F
4626:4
4618:2
4614:x
4610:5
4590:4
4587:+
4582:2
4578:x
4574:5
4560:x
4539:.
4534:1
4531:+
4528:n
4524:F
4515:2
4512:+
4509:n
4505:F
4501:=
4496:n
4492:F
4476:n
4474:F
4468:n
4451:.
4446:1
4440:n
4436:F
4432:+
4424:n
4420:F
4416:=
4411:n
4386:1
4383:+
4377:=
4372:2
4343:/
4339:1
4333:=
4310:.
4305:n
4301:F
4297:+
4289:1
4286:+
4283:n
4279:F
4275:=
4270:n
4266:F
4262:+
4256:)
4251:1
4245:n
4241:F
4237:+
4232:n
4228:F
4224:(
4221:=
4213:1
4207:n
4203:F
4199:+
4196:)
4193:1
4190:+
4184:(
4179:n
4175:F
4171:=
4163:1
4157:n
4153:F
4149:+
4144:2
4134:n
4130:F
4126:=
4120:)
4115:1
4109:n
4105:F
4101:+
4093:n
4089:F
4085:(
4082:=
4077:1
4074:+
4071:n
4054:n
4031:.
4026:1
4020:n
4016:F
4012:+
4004:n
4000:F
3996:=
3991:n
3936:n
3909:,
3906:1
3903:+
3897:=
3892:2
3825:m
3817:=
3810:n
3806:F
3800:m
3797:+
3794:n
3790:F
3776:n
3740:/
3734:0
3730:U
3723:=
3718:1
3714:U
3691:1
3687:U
3664:0
3660:U
3637:.
3631:=
3624:n
3620:F
3614:1
3611:+
3608:n
3604:F
3590:n
3529:.
3523:b
3503:n
3497:=
3486:b
3478:n
3466:n
3462:F
3457:b
3447:d
3441:d
3426:n
3400:n
3388:n
3384:F
3367:5
3361:/
3355:n
3335:n
3333:F
3305:=
3302:)
3296:(
3257:=
3254:)
3248:(
3222:)
3216:(
3199:/
3195:)
3192:x
3189:(
3173:=
3170:)
3164:(
3154:/
3150:)
3147:x
3144:(
3135:=
3132:)
3129:x
3126:(
3093:,
3090:0
3084:F
3078:,
3070:)
3067:2
3063:/
3059:1
3056:+
3053:F
3050:(
3045:5
3022:=
3019:)
3016:F
3013:(
3007:t
3004:s
3001:e
2998:g
2995:r
2992:a
2989:l
2984:n
2973:F
2947:F
2941:,
2933:F
2928:5
2905:=
2902:)
2899:F
2896:(
2893:n
2883:F
2877:n
2870:n
2865:n
2843:n
2837:,
2827:5
2821:n
2807:=
2802:n
2798:F
2770:5
2764:n
2741:n
2737:F
2730:n
2713:2
2710:1
2701:|
2695:5
2689:n
2679:|
2647:.
2641:5
2634:1
2630:U
2618:0
2614:U
2607:=
2600:b
2593:,
2587:5
2577:0
2573:U
2564:1
2560:U
2553:=
2546:a
2516:n
2508:b
2505:+
2500:n
2492:a
2489:=
2484:n
2480:U
2466:1
2463:U
2457:0
2454:U
2434:,
2431:a
2425:=
2422:b
2418:,
2412:5
2408:1
2403:=
2387:1
2382:=
2379:a
2348:1
2345:=
2338:b
2332:+
2329:a
2319:0
2316:=
2309:b
2306:+
2303:a
2296:{
2283:b
2279:a
2273:n
2269:U
2262:1
2259:U
2252:0
2249:U
2244:b
2240:a
2220:.
2215:2
2209:n
2205:U
2201:+
2196:1
2190:n
2186:U
2182:=
2170:2
2164:n
2156:b
2153:+
2148:2
2142:n
2134:a
2131:+
2126:1
2120:n
2112:b
2109:+
2104:1
2098:n
2090:a
2087:=
2077:)
2072:2
2066:n
2058:+
2053:1
2047:n
2039:(
2036:b
2033:+
2030:)
2025:2
2019:n
2011:+
2006:1
2000:n
1992:(
1989:a
1986:=
1974:n
1966:b
1963:+
1958:n
1950:a
1947:=
1938:n
1934:U
1903:n
1895:b
1892:+
1887:n
1879:a
1876:=
1871:n
1867:U
1854:b
1850:a
1830:.
1825:2
1819:n
1811:+
1806:1
1800:n
1792:=
1783:n
1771:,
1766:2
1760:n
1752:+
1747:1
1741:n
1733:=
1724:n
1703:Ï
1699:Ï
1685:,
1680:2
1674:n
1670:x
1666:+
1661:1
1655:n
1651:x
1647:=
1642:n
1638:x
1617:1
1614:+
1611:x
1608:=
1603:2
1599:x
1588:Ï
1584:Ï
1568:.
1562:1
1553:2
1546:n
1539:)
1529:(
1521:n
1510:=
1504:5
1497:n
1490:)
1480:(
1472:n
1461:=
1456:n
1452:F
1427:1
1413:=
1388:.
1364:1
1356:=
1347:1
1344:=
1339:2
1333:5
1325:1
1319:=
1300:Ï
1265:2
1259:5
1254:+
1251:1
1245:=
1218:,
1212:5
1205:n
1192:n
1181:=
1162:n
1149:n
1138:=
1133:n
1129:F
1066:F
1040:n
1036:n
1030:n
1023:n
1018:n
978:(
929:.
902:m
898:F
891:m
887:F
879:m
875:F
870:m
860:(
848:m
844:F
839:m
822:5
819:F
813:6
810:F
804:7
801:F
703:F
693:F
683:F
673:F
663:F
653:F
643:F
633:F
623:F
613:F
606:9
603:F
596:8
593:F
586:7
583:F
576:6
573:F
566:5
563:F
556:4
553:F
546:3
543:F
536:2
533:F
526:1
523:F
516:0
513:F
499:n
497:F
487:n
470:2
464:n
460:F
456:+
451:1
445:n
441:F
437:=
432:n
428:F
407:,
404:1
401:=
396:2
392:F
388:=
383:1
379:F
358:0
355:=
350:0
346:F
330:n
313:2
307:n
303:F
299:+
294:1
288:n
284:F
280:=
275:n
271:F
250:,
247:1
244:=
239:1
235:F
230:,
227:0
224:=
219:0
215:F
165:n
161:n
157:n
44:n
42:F
23:.
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