28:
20:
3426:
3414:
3402:
3390:
3378:
3366:
3354:
3342:
3330:
3318:
3306:
3294:
3282:
3270:
3258:
3246:
3234:
3222:
3210:
3198:
3186:
3174:
3162:
3150:
3138:
3126:
3114:
3102:
3090:
3078:
3066:
3054:
3042:
3030:
3018:
3006:
2994:
2982:
2970:
2958:
2946:
2934:
2922:
2910:
2898:
2886:
2874:
2862:
2850:
2838:
2826:
2814:
2802:
2790:
2778:
2766:
2754:
2742:
2730:
2718:
2706:
2694:
2682:
2670:
2658:
2646:
2634:
2622:
2610:
2598:
2586:
2574:
2562:
2550:
2538:
2526:
2514:
2502:
2490:
2478:
2466:
2454:
2442:
2430:
2418:
2406:
2394:
2382:
2370:
2358:
4554:
section between two walls so the automaton must eventually start repeating inside each section, though the period may be very long if the section is wide enough. These walls will form with probability 1 for completely random initial conditions. However, if the condition is added that the lengths of runs of consecutive 0s or 1s must always be odd, then the automaton displays Class 3 behavior since the walls can never form.
2346:
2334:
2322:
2310:
2298:
2286:
2274:
2262:
2250:
2238:
1296:
77:, to assign each rule a number from 0 to 255 which has become standard. Each possible current configuration is written in order, 111, 110, ..., 001, 000, and the resulting state for each of these configurations is written in the same order and interpreted as the binary representation of an integer. This number is taken to be the rule number of the automaton. For example, 110
1084:
4553:
Rule 73 is Class 2 because any time there are two consecutive 1s surrounded by 0s, this feature is preserved in succeeding generations. This effectively creates walls which block the flow of information between different parts of the array. There are a finite number of possible configurations in the
2230:
These images depict space-time diagrams, in which each row of pixels shows the cells of the automaton at a single point in time, with time increasing downwards. They start with an initial automaton state in which a single cell, the pixel in the center of the top row of pixels, is in state 1 and all
3470:
In the following gallery, this evolution from random initial conditions is shown for each of the 88 inequivalent rules. Below each image is the rule number used to produce the image, and in brackets the rule numbers of equivalent rules produced by reflection or complementing are included, if they
4549:
In some cases the behavior of a cellular automaton is not immediately obvious. For example, for Rule 62, interacting structures develop as in a Class 4. But in these interactions at least one of the structures is annihilated so the automaton eventually enters a repetitive state and the cellular
168:
Although there are 256 possible rules, many of these are trivially equivalent to each other up to a simple transformation of the underlying geometry. The first such transformation is reflection through a vertical axis and the result of applying this transformation to a given rule is called the
406:
Finally, the previous two transformations can be applied successively to a rule to obtain the mirrored complementary rule. For example, the mirrored complementary rule of rule 110 is rule 193. There are 16 rules which are the same as their mirrored complementary rules.
425:
One method used to study these automata is to follow its history with an initial state of all 0s except for a single cell with a 1. When the rule number is even (so that an input of 000 does not compute to a 1) it makes sense to interpret state at each time,
3441:
A second way to investigate the behavior of these automata is to examine its history starting with a random state. This behavior can be better understood in terms of
Wolfram classes. Wolfram gives the following examples as typical rules of each class.
68:
There are 8 = 2 possible configurations for a cell and its two immediate neighbors. The rule defining the cellular automaton must specify the resulting state for each of these possibilities so there are 256 = 2 possible elementary cellular automata.
1291:{\displaystyle a(t)={\begin{cases}1,&{\mbox{if }}t=0\\7,&{\mbox{if }}t=1\\{\dfrac {1+5\cdot 4^{n}}{3}},&{\mbox{if }}t{\mbox{ is even otherwise}}\\{\dfrac {10+11\cdot 4^{n}}{6}},&{\mbox{if }}t{\mbox{ is odd otherwise}}\end{cases}}}
5044:
51:
where there are two possible states (labeled 0 and 1) and the rule to determine the state of a cell in the next generation depends only on the current state of the cell and its two immediate neighbors. There is an elementary cellular automaton
4557:
Rule 54 is Class 4 and also appears to be capable of universal computation, but has not been studied as thoroughly as Rule 110. Many interacting structures have been cataloged which collectively are expected to be sufficient for universality.
1811:
1003:
1427:
1641:
4471:
4447:
4423:
4387:
4375:
4327:
4315:
4303:
4291:
4267:
4216:
4192:
4057:
4045:
4033:
3961:
3937:
3913:
4129:
4093:
4009:
3901:
3889:
3877:
3829:
3790:
3766:
3742:
3718:
3646:
3622:
3598:
3949:
3841:
3730:
3610:
3754:
3634:
3586:
3574:
3550:
3502:
3514:
3562:
4510:
4495:
4459:
4435:
4411:
4399:
4351:
4339:
4279:
4255:
4243:
4231:
4204:
4168:
621:
4156:
4141:
4105:
4081:
4069:
3997:
3973:
3925:
3853:
3817:
3805:
3694:
3670:
4021:
3865:
3778:
3682:
3658:
883:
3526:
3490:
3478:
3538:
4534:
4522:
4483:
4363:
4180:
774:
4117:
3985:
3706:
1910:
2155:
535:
1995:
704:
2216:
2056:
3455:
Class 4: Cellular automata which form areas of repetitive or stable states, but also form structures that interact with each other in complicated ways. An example is
4944:
Castillo-Ramirez, A., Gadouleau, M. (2020), Elementary, Finite and Linear vN-Regular
Cellular Automata, Information and Computation, vol. 274, 104533. Section 3.
3471:
exist. As mentioned above, the reflected rule would produce a reflected image, while the complementary rule would produce an image with black and white swapped.
269:
The second such transformation is to exchange the roles of 0 and 1 in the definition. The result of applying this transformation to a given rule is called the
1668:
895:
1308:
1530:
173:. These rules will exhibit the same behavior up to reflection through a vertical axis, and so are equivalent in a computational sense.
2077:
1928:
1829:
1659:
1521:
1495:
1469:
1445:
1075:
1053:
1021:
795:
641:
457:
1060:
modulo 2 and interpreting them as integers in base 4. Note that rules 18, 26, 82, 146, 154, 210 and 218 generate the same sequence.
3467:
Each computed result is placed under that result's source creating a two-dimensional representation of the system's evolution.
40:
547:
176:
For example, if the definition of rule 110 is reflected through a vertical line, the following rule (rule 124) is obtained:
5186:
3449:
Class 2: Cellular automata which rapidly converge to a repetitive or stable state. Examples are rules 4, 108, 218 and 250.
804:
5181:
5037:
23:
An animation of how an evolution is determined in Rule 30, one of the 256 possible rules of elementary cellular automata.
1108:
716:
5057:
1838:
1448:). This is simply the sequence generated by rule 60 (which is its mirror rule) multiplied by successive powers of 2.
2090:
470:
3452:
Class 3: Cellular automata which appear to remain in a random state. Examples are rules 22, 30, 126, 150, 182.
1941:
650:
3446:
Class 1: Cellular automata which rapidly converge to a uniform state. Examples are rules 0, 32, 160 and 232.
410:
Of the 256 elementary cellular automata, there are 88 which are inequivalent under these transformations.
2167:
2007:
5201:
4920:
4572:
1028:
modulo 2 and interpreting them as integers in binary, which can be graphically represented by a
27:
5112:"Phenomenology of glider collisions in cellular automaton Rule 54 and associated logical gates"
4470:
4446:
4422:
4386:
4374:
4326:
4314:
4302:
4290:
4266:
4215:
4191:
1462:
The sequence generated is 1, 6, 28, 104, 496, 1568, 7360, 27520, 130304, 396800, ... (sequence
4056:
4044:
4032:
3960:
3936:
3912:
3460:
1477:
4128:
4092:
4008:
3900:
3888:
3876:
3828:
3789:
3765:
3741:
3717:
3645:
3621:
3597:
438:) of integers. In many cases these sequences have simple, closed form expressions or have a
5126:
3948:
3840:
3729:
3609:
1057:
1025:
3753:
3633:
3585:
3573:
3549:
3501:
8:
3513:
1029:
439:
57:
5130:
3561:
273:. For example, if this transformation is applied to rule 110, we get the following rule
5107:
4971:
461:
417:
of the monoid of one-dimensional cellular automata, as they both preserve composition.
48:
5142:
5111:
5053:
4991:
4892:
4873:
4854:
4835:
4816:
4797:
4778:
4759:
4740:
4721:
4702:
4683:
4664:
4645:
4626:
4607:
4588:
4569:
4509:
4494:
4458:
4434:
4410:
4398:
4350:
4338:
4278:
4254:
4242:
4230:
4203:
4167:
5134:
4981:
4155:
4140:
4104:
4080:
4068:
3996:
3972:
3924:
3852:
3816:
3804:
3693:
3669:
2081:
1932:
5021:
4020:
3864:
3777:
3681:
3657:
1498:). This can be obtained by taking the coefficients of the successive powers of (1+
19:
5094:
5083:
5072:
5033:
3525:
3489:
3477:
1806:{\displaystyle {\frac {1+3x+4x^{2}+12x^{3}+8x^{4}-8x^{5}}{(1-x^{2})(1-16x^{4})}}}
1473:
70:
5138:
3537:
3425:
3413:
3401:
3389:
3377:
3365:
3353:
3341:
3329:
3317:
3305:
3293:
3281:
3269:
3257:
3245:
3233:
3221:
3209:
3197:
3185:
3173:
3161:
3149:
3137:
3125:
3113:
3101:
3089:
3077:
3065:
3053:
3041:
3029:
3017:
3005:
2993:
2981:
2969:
2957:
2945:
2933:
2921:
2909:
2897:
2885:
2873:
2861:
2849:
2837:
2825:
2813:
2801:
2789:
2777:
2765:
2753:
2741:
2729:
2717:
2705:
2693:
2681:
2669:
2657:
2645:
2633:
2621:
2609:
2597:
2585:
2573:
2561:
2549:
2537:
2525:
2513:
2501:
2489:
2477:
2465:
2453:
2441:
2429:
2417:
2405:
2393:
2381:
2369:
2357:
780:
Note that rules 58, 114, 122, 178, 186, 242 and 250 generate the same sequence.
4533:
4521:
4482:
4362:
4179:
2070:
The sequence generated is 1, 7, 31, 127, 511, 2047, 8191, 32767, ... (sequence
1822:
The sequence generated is 1, 7, 29, 119, 477, 1911, 7645, 30583, ... (sequence
1514:
The sequence generated is 1, 7, 29, 115, 477, 1843, 7645, 29491, ... (sequence
1488:
The sequence generated is 1, 7, 21, 107, 273, 1911, 5189, 28123, ... (sequence
1438:
The sequence generated is 1, 6, 20, 120, 272, 1632, 5440, 32640, ... (sequence
1068:
The sequence generated is 1, 7, 27, 119, 427, 1879, 6827, 30039, ... (sequence
788:
The sequence generated is 1, 7, 17, 119, 273, 1911, 4369, 30583, ... (sequence
2345:
2333:
2321:
2309:
2297:
2285:
2273:
2261:
2249:
2237:
1046:
The sequence generated is 1, 5, 17, 85, 257, 1285, 4369, 21845, ... (sequence
634:
The sequence generated is 1, 5, 21, 85, 341, 1365, 5461, 21845, ... (sequence
5195:
5146:
4995:
4895:
4876:
4857:
4838:
4819:
4800:
4781:
4762:
4743:
4724:
4116:
3984:
3705:
414:
5171:
4705:
4686:
4667:
4648:
4629:
4610:
4591:
4945:
74:
5187:
Minimal CA emulation with
Wolfram rule parser online in vanilla Javascript
31:
All the 256 elementary cellular automaton rules (click or tap to enlarge).
36:
60:, and as such it is one of the simplest possible models of computation.
1921:
The sequence generated is 1, 3, 7, 15, 31, 63, 127, 255, ... (sequence
4986:
4959:
4567:
1652:
The sequence generated is 1, 3, 5, 15, 29, 55, 93, 247, ... (sequence
998:{\displaystyle a(t)={\frac {22\cdot 4^{t}-6(-4)^{t}-4+3(-1)^{t}}{15}}}
450:
The sequence generated is 1, 3, 5, 11, 21, 43, 85, 171, ... (sequence
4900:
4881:
4862:
4843:
4824:
4805:
4786:
4767:
4748:
4729:
4710:
4691:
4672:
4653:
4634:
4615:
4596:
4577:
1014:
The sequence generated is 1, 3, 5, 15, 17, 51, 85, 255, ...(sequence
1422:{\displaystyle {\frac {(1+2x)(1+5x-16x^{4})}{(1-x^{2})(1-16x^{2})}}}
403:
There are 16 rules which are the same as their complementary rules.
5175:
4921:"Elementary cellular automaton in the Fōrmulæ programming language"
4500:
4221:
3456:
1636:{\displaystyle {\frac {1+7x+12x^{2}-4x^{3}}{(1-x^{2})(1-16x^{2})}}}
1457:
53:
4976:
5163:
4146:
3795:
1041:
266:. Of the 256 elementary cellular automata, 64 are amphichiral.
5172:
32 bytes long MS-DOS executable drawing by cellular automaton
2072:
1923:
1824:
1654:
1516:
1490:
1472:). Rule 110 has the perhaps surprising property that it is
1464:
1440:
1284:
1070:
1048:
1016:
790:
636:
452:
262:
Rules which are the same as their mirrored rule are called
430:, as an integer expressed in binary, producing a sequence
338:
and, after reordering, we discover that this is rule 137:
5105:
1506:) modulo 2 and interpreting them as integers in binary.
616:{\displaystyle a(t)={\frac {4\cdot 2^{t}-(-1)^{t}}{3}}}
4964:
1275:
1265:
1216:
1206:
1148:
1120:
2170:
2093:
2010:
1944:
1841:
1671:
1533:
1311:
1227:
1168:
1087:
1056:). This can be obtained by taking successive rows of
1024:). This can be obtained by taking successive rows of
898:
807:
719:
653:
550:
473:
442:
with a simple form. The following rules are notable:
413:
It turns out that reflection and complementation are
878:{\displaystyle {\frac {1+7x}{(1-x^{2})(1-16x^{2})}}}
5046:Cellular Automata and Complexity: Collected Papers
2210:
2149:
2050:
1989:
1904:
1805:
1635:
1421:
1290:
997:
877:
768:
698:
615:
529:
4890:
4871:
4852:
4833:
4814:
4795:
4776:
4757:
4738:
4719:
85:. So rule 110 is defined by the transition rule:
5193:
4700:
4681:
4662:
4643:
4624:
4605:
4586:
2222:Note that rule 254 generates the same sequence.
2080:). This is every other entry in the sequence of
769:{\displaystyle a(t)={\frac {4\cdot 4^{t}-1}{3}}}
1905:{\displaystyle {\frac {1+3x}{(1-x^{2})(1-4x)}}}
163:
5182:A showcase of all the rules picked at random
5106:Martínez, Genaro Juárez; Adamatzky, Andrew;
2062:Note: rule 252 generates the same sequence.
3459:. Rule 110 has been shown to be capable of
2150:{\displaystyle {\frac {1+2x}{(1-x)(1-4x)}}}
530:{\displaystyle {\frac {1+2x}{(1+x)(1-2x)}}}
5038:"Tables of Cellular Automaton Properties"
4985:
4975:
4960:"A Concrete View of Rule 110 Computation"
4946:https://doi.org/10.1016/j.ic.2020.104533
2225:
1990:{\displaystyle {\frac {1}{(1-x)(1-2x)}}}
699:{\displaystyle {\frac {1}{(1-x)(1-4x)}}}
26:
18:
5032:
4912:
3436:
63:
5194:
5164:"Elementary Cellular Automata" at the
626:Rule 156 generates the same sequence.
4891:
4872:
4853:
4834:
4815:
4796:
4777:
4758:
4739:
4720:
4701:
4682:
4663:
4644:
4625:
4606:
4587:
4568:
420:
56:, defined below) which is capable of
5052:. Westview Press. pp. 516–521.
4957:
4918:
2211:{\displaystyle a(t)=2\cdot 4^{t}-1}
2051:{\displaystyle a(t)=2\cdot 2^{t}-1}
13:
2161:It has the closed form expression
2001:It has the closed form expression
889:It has the closed form expression
710:It has the closed form expression
541:It has the closed form expression
14:
5213:
5157:
5166:Wolfram Atlas of Simple Programs
4544:
4532:
4520:
4508:
4493:
4481:
4469:
4457:
4445:
4433:
4421:
4409:
4397:
4385:
4373:
4361:
4349:
4337:
4325:
4313:
4301:
4289:
4277:
4265:
4253:
4241:
4229:
4214:
4202:
4190:
4178:
4166:
4154:
4139:
4127:
4115:
4103:
4091:
4079:
4067:
4055:
4043:
4031:
4019:
4007:
3995:
3983:
3971:
3959:
3947:
3935:
3923:
3911:
3899:
3887:
3875:
3863:
3851:
3839:
3827:
3815:
3803:
3788:
3776:
3764:
3752:
3740:
3728:
3716:
3704:
3692:
3680:
3668:
3656:
3644:
3632:
3620:
3608:
3596:
3584:
3572:
3560:
3548:
3536:
3524:
3512:
3500:
3488:
3476:
3424:
3412:
3400:
3388:
3376:
3364:
3352:
3340:
3328:
3316:
3304:
3292:
3280:
3268:
3256:
3244:
3232:
3220:
3208:
3196:
3184:
3172:
3160:
3148:
3136:
3124:
3112:
3100:
3088:
3076:
3064:
3052:
3040:
3028:
3016:
3004:
2992:
2980:
2968:
2956:
2944:
2932:
2920:
2908:
2896:
2884:
2872:
2860:
2848:
2836:
2824:
2812:
2800:
2788:
2776:
2764:
2752:
2740:
2728:
2716:
2704:
2692:
2680:
2668:
2656:
2644:
2632:
2620:
2608:
2596:
2584:
2572:
2560:
2548:
2536:
2524:
2512:
2500:
2488:
2476:
2464:
2452:
2440:
2428:
2416:
2404:
2392:
2380:
2368:
2356:
2344:
2332:
2320:
2308:
2296:
2284:
2272:
2260:
2248:
2236:
1832:). This has generating function
1662:). This has generating function
1524:). This has generating function
798:). This has generating function
644:). This has generating function
73:proposed a scheme, known as the
5099:
4573:"Elementary Cellular Automaton"
5119:Chaos, Solitons & Fractals
5088:
5077:
5066:
5026:
5015:
5002:
4951:
4938:
2180:
2174:
2141:
2126:
2123:
2111:
2020:
2014:
1981:
1966:
1963:
1951:
1896:
1881:
1878:
1859:
1797:
1775:
1772:
1753:
1627:
1605:
1602:
1583:
1413:
1391:
1388:
1369:
1364:
1333:
1330:
1315:
1097:
1091:
980:
970:
949:
939:
908:
902:
869:
847:
844:
825:
729:
723:
690:
675:
672:
660:
598:
588:
560:
554:
521:
506:
503:
491:
1:
4561:
1302:This has generating function
45:elementary cellular automaton
4958:Cook, Matthew (2009-06-25).
2084:and has generating function
1935:and has generating function
1078:). This can be expressed as
464:and has generating function
7:
5139:10.1016/j.chaos.2005.05.013
2065:
1931:). This is the sequence of
1916:
1817:
1647:
1509:
1483:
1451:
1433:
460:). This is the sequence of
164:Reflections and complements
10:
5218:
1455:
1063:
1039:
1035:
1009:
783:
629:
445:
373:new state for center cell
308:new state for center cell
238:new state for center cell
147:new state for center cell
5022:Rule 110 - Wolfram|Alpha
4476:Rule 172 (202, 216, 228)
4452:Rule 168 (224, 234, 248)
4428:Rule 162 (176, 186, 242)
4392:Rule 154 (166, 180, 210)
4380:Rule 152 (188, 194, 230)
4332:Rule 140 (196, 206, 220)
4320:Rule 138 (174, 208, 244)
4308:Rule 136 (192, 238, 252)
4296:Rule 134 (148, 158, 214)
4272:Rule 130 (144, 190, 246)
4197:Rule 106 (120, 169, 225)
5095:Rule 54 - Wolfram|Alpha
5084:Rule 73 - Wolfram|Alpha
5073:Rule 62 - Wolfram|Alpha
4062:Rule 62 (118, 131, 145)
4050:Rule 60 (102, 153, 195)
4038:Rule 58 (114, 163, 177)
3966:Rule 46 (116, 139, 209)
3942:Rule 44 (100, 203, 217)
3918:Rule 42 (112, 171, 241)
1218: is even otherwise
4550:automaton is Class 2.
4134:Rule 78 (92, 141, 197)
4098:Rule 74 (88, 173, 229)
4014:Rule 56 (98, 185, 227)
3906:Rule 41 (97, 107, 121)
3894:Rule 40 (96, 235, 249)
3882:Rule 38 (52, 155, 211)
3834:Rule 34 (48, 187, 243)
3771:Rule 28 (70, 157, 199)
3747:Rule 26 (82, 167, 181)
3723:Rule 24 (66, 189, 231)
3651:Rule 14 (84, 143, 213)
3627:Rule 12 (68, 207, 221)
3603:Rule 10 (80, 175, 245)
2212:
2151:
2052:
1991:
1906:
1807:
1637:
1476:, and thus capable of
1423:
1292:
1277: is odd otherwise
999:
879:
770:
700:
617:
531:
32:
24:
5010:A New Kind of Science
3954:Rule 45 (75, 89, 101)
3846:Rule 35 (49, 59, 115)
3735:Rule 25 (61, 67, 103)
3615:Rule 11 (47, 81, 117)
3591:Rule 9 (65, 111, 125)
3579:Rule 8 (64, 239, 253)
3555:Rule 6 (20, 159, 215)
3507:Rule 2 (16, 191, 247)
3461:universal computation
2226:Images for rules 0-99
2213:
2152:
2053:
1992:
1907:
1808:
1638:
1478:universal computation
1424:
1293:
1000:
880:
771:
701:
618:
532:
58:universal computation
47:is a one-dimensional
30:
22:
4919:R.Ugalde, Laurence.
3759:Rule 27 (39, 53, 83)
3639:Rule 13 (69, 79, 93)
3519:Rule 3 (17, 63, 119)
3437:Random initial state
2168:
2091:
2008:
1942:
1839:
1669:
1531:
1309:
1085:
896:
805:
717:
651:
548:
471:
64:The numbering system
41:computability theory
5131:2006CSF....28..100M
5108:McIntosh, Harold V.
3567:Rule 7 (21, 31, 87)
2231:other cells are 0.
1030:Sierpinski triangle
440:generating function
256:=(L+C+L*C+L*C*R)%2
157:=(C+R+C*R+L*C*R)%2
16:Mathematics concept
4893:Weisstein, Eric W.
4874:Weisstein, Eric W.
4855:Weisstein, Eric W.
4836:Weisstein, Eric W.
4817:Weisstein, Eric W.
4798:Weisstein, Eric W.
4779:Weisstein, Eric W.
4760:Weisstein, Eric W.
4741:Weisstein, Eric W.
4722:Weisstein, Eric W.
4703:Weisstein, Eric W.
4684:Weisstein, Eric W.
4665:Weisstein, Eric W.
4646:Weisstein, Eric W.
4627:Weisstein, Eric W.
4608:Weisstein, Eric W.
4589:Weisstein, Eric W.
4570:Weisstein, Eric W.
2208:
2147:
2048:
1987:
1902:
1803:
1633:
1419:
1288:
1283:
1279:
1269:
1257:
1220:
1210:
1198:
1152:
1124:
995:
875:
766:
696:
613:
527:
462:Jacobsthal numbers
421:Single 1 histories
271:complementary rule
49:cellular automaton
33:
25:
5202:Cellular automata
5008:Stephen Wolfram,
4987:10.4204/EPTCS.1.4
2145:
1985:
1900:
1801:
1631:
1417:
1278:
1268:
1256:
1219:
1209:
1197:
1151:
1123:
1058:Pascal's triangle
1026:Pascal's triangle
993:
873:
764:
694:
611:
525:
401:
400:
336:
335:
260:
259:
161:
160:
5209:
5151:
5150:
5116:
5103:
5097:
5092:
5086:
5081:
5075:
5070:
5064:
5063:
5051:
5042:
5034:Wolfram, Stephen
5030:
5024:
5019:
5013:
5006:
5000:
4999:
4989:
4979:
4955:
4949:
4942:
4936:
4935:
4933:
4931:
4916:
4906:
4905:
4887:
4886:
4868:
4867:
4849:
4848:
4830:
4829:
4811:
4810:
4792:
4791:
4773:
4772:
4754:
4753:
4735:
4734:
4716:
4715:
4697:
4696:
4678:
4677:
4659:
4658:
4640:
4639:
4621:
4620:
4602:
4601:
4583:
4582:
4536:
4524:
4512:
4497:
4485:
4473:
4461:
4449:
4437:
4425:
4413:
4401:
4389:
4377:
4365:
4353:
4341:
4329:
4317:
4305:
4293:
4281:
4269:
4257:
4245:
4233:
4218:
4206:
4194:
4182:
4170:
4158:
4143:
4131:
4119:
4107:
4095:
4083:
4071:
4059:
4047:
4035:
4023:
4011:
3999:
3987:
3975:
3963:
3951:
3939:
3927:
3915:
3903:
3891:
3879:
3867:
3855:
3843:
3831:
3819:
3807:
3792:
3780:
3768:
3756:
3744:
3732:
3720:
3708:
3696:
3684:
3672:
3660:
3648:
3636:
3624:
3612:
3600:
3588:
3576:
3564:
3552:
3540:
3528:
3516:
3504:
3492:
3480:
3428:
3416:
3404:
3392:
3380:
3368:
3356:
3344:
3332:
3320:
3308:
3296:
3284:
3272:
3260:
3248:
3236:
3224:
3212:
3200:
3188:
3176:
3164:
3152:
3140:
3128:
3116:
3104:
3092:
3080:
3068:
3056:
3044:
3032:
3020:
3008:
2996:
2984:
2972:
2960:
2948:
2936:
2924:
2912:
2900:
2888:
2876:
2864:
2852:
2840:
2828:
2816:
2804:
2792:
2780:
2768:
2756:
2744:
2732:
2720:
2708:
2696:
2684:
2672:
2660:
2648:
2636:
2624:
2612:
2600:
2588:
2576:
2564:
2552:
2540:
2528:
2516:
2504:
2492:
2480:
2468:
2456:
2444:
2432:
2420:
2408:
2396:
2384:
2372:
2360:
2348:
2336:
2324:
2312:
2300:
2288:
2276:
2264:
2252:
2240:
2217:
2215:
2214:
2209:
2201:
2200:
2156:
2154:
2153:
2148:
2146:
2144:
2109:
2095:
2082:Mersenne numbers
2075:
2057:
2055:
2054:
2049:
2041:
2040:
1996:
1994:
1993:
1988:
1986:
1984:
1946:
1933:Mersenne numbers
1926:
1911:
1909:
1908:
1903:
1901:
1899:
1877:
1876:
1857:
1843:
1827:
1812:
1810:
1809:
1804:
1802:
1800:
1796:
1795:
1771:
1770:
1751:
1750:
1749:
1734:
1733:
1718:
1717:
1702:
1701:
1673:
1657:
1642:
1640:
1639:
1634:
1632:
1630:
1626:
1625:
1601:
1600:
1581:
1580:
1579:
1564:
1563:
1535:
1519:
1493:
1467:
1443:
1428:
1426:
1425:
1420:
1418:
1416:
1412:
1411:
1387:
1386:
1367:
1363:
1362:
1313:
1297:
1295:
1294:
1289:
1287:
1286:
1280:
1276:
1270:
1266:
1258:
1252:
1251:
1250:
1228:
1221:
1217:
1211:
1207:
1199:
1193:
1192:
1191:
1169:
1153:
1149:
1125:
1121:
1073:
1051:
1019:
1004:
1002:
1001:
996:
994:
989:
988:
987:
957:
956:
932:
931:
915:
884:
882:
881:
876:
874:
872:
868:
867:
843:
842:
823:
809:
793:
775:
773:
772:
767:
765:
760:
753:
752:
736:
705:
703:
702:
697:
695:
693:
655:
639:
622:
620:
619:
614:
612:
607:
606:
605:
584:
583:
567:
536:
534:
533:
528:
526:
524:
489:
475:
455:
344:current pattern
341:
340:
279:current pattern
276:
275:
206:current pattern
179:
178:
115:current pattern
88:
87:
5217:
5216:
5212:
5211:
5210:
5208:
5207:
5206:
5192:
5191:
5160:
5155:
5154:
5114:
5104:
5100:
5093:
5089:
5082:
5078:
5071:
5067:
5060:
5049:
5040:
5031:
5027:
5020:
5016:
5007:
5003:
4956:
4952:
4943:
4939:
4929:
4927:
4917:
4913:
4564:
4547:
4540:
4537:
4528:
4525:
4516:
4513:
4504:
4498:
4489:
4486:
4477:
4474:
4465:
4462:
4453:
4450:
4441:
4438:
4429:
4426:
4417:
4414:
4405:
4402:
4393:
4390:
4381:
4378:
4369:
4366:
4357:
4354:
4345:
4342:
4333:
4330:
4321:
4318:
4309:
4306:
4297:
4294:
4285:
4282:
4273:
4270:
4261:
4258:
4249:
4246:
4237:
4234:
4225:
4224:(124, 137, 193)
4219:
4210:
4207:
4198:
4195:
4186:
4183:
4174:
4171:
4162:
4159:
4150:
4144:
4135:
4132:
4123:
4120:
4111:
4108:
4099:
4096:
4087:
4084:
4075:
4072:
4063:
4060:
4051:
4048:
4039:
4036:
4027:
4024:
4015:
4012:
4003:
4000:
3991:
3988:
3979:
3976:
3967:
3964:
3955:
3952:
3943:
3940:
3931:
3928:
3919:
3916:
3907:
3904:
3895:
3892:
3883:
3880:
3871:
3868:
3859:
3856:
3847:
3844:
3835:
3832:
3823:
3820:
3811:
3808:
3799:
3793:
3784:
3781:
3772:
3769:
3760:
3757:
3748:
3745:
3736:
3733:
3724:
3721:
3712:
3709:
3700:
3697:
3688:
3685:
3676:
3673:
3664:
3661:
3652:
3649:
3640:
3637:
3628:
3625:
3616:
3613:
3604:
3601:
3592:
3589:
3580:
3577:
3568:
3565:
3556:
3553:
3544:
3541:
3532:
3529:
3520:
3517:
3508:
3505:
3496:
3493:
3484:
3481:
3439:
3432:
3429:
3420:
3417:
3408:
3405:
3396:
3393:
3384:
3381:
3372:
3369:
3360:
3357:
3348:
3345:
3336:
3333:
3324:
3321:
3312:
3309:
3300:
3297:
3288:
3285:
3276:
3273:
3264:
3261:
3252:
3249:
3240:
3237:
3228:
3225:
3216:
3213:
3204:
3201:
3192:
3189:
3180:
3177:
3168:
3165:
3156:
3153:
3144:
3141:
3132:
3129:
3120:
3117:
3108:
3105:
3096:
3093:
3084:
3081:
3072:
3069:
3060:
3057:
3048:
3045:
3036:
3033:
3024:
3021:
3012:
3009:
3000:
2997:
2988:
2985:
2976:
2973:
2964:
2961:
2952:
2949:
2940:
2937:
2928:
2925:
2916:
2913:
2904:
2901:
2892:
2889:
2880:
2877:
2868:
2865:
2856:
2853:
2844:
2841:
2832:
2829:
2820:
2817:
2808:
2805:
2796:
2793:
2784:
2781:
2772:
2769:
2760:
2757:
2748:
2745:
2736:
2733:
2724:
2721:
2712:
2709:
2700:
2697:
2688:
2685:
2676:
2673:
2664:
2661:
2652:
2649:
2640:
2637:
2628:
2625:
2616:
2613:
2604:
2601:
2592:
2589:
2580:
2577:
2568:
2565:
2556:
2553:
2544:
2541:
2532:
2529:
2520:
2517:
2508:
2505:
2496:
2493:
2484:
2481:
2472:
2469:
2460:
2457:
2448:
2445:
2436:
2433:
2424:
2421:
2412:
2409:
2400:
2397:
2388:
2385:
2376:
2373:
2364:
2361:
2352:
2349:
2340:
2337:
2328:
2325:
2316:
2313:
2304:
2301:
2292:
2289:
2280:
2277:
2268:
2265:
2256:
2253:
2244:
2241:
2228:
2196:
2192:
2169:
2166:
2165:
2110:
2096:
2094:
2092:
2089:
2088:
2071:
2068:
2036:
2032:
2009:
2006:
2005:
1950:
1945:
1943:
1940:
1939:
1922:
1919:
1872:
1868:
1858:
1844:
1842:
1840:
1837:
1836:
1823:
1820:
1791:
1787:
1766:
1762:
1752:
1745:
1741:
1729:
1725:
1713:
1709:
1697:
1693:
1674:
1672:
1670:
1667:
1666:
1653:
1650:
1621:
1617:
1596:
1592:
1582:
1575:
1571:
1559:
1555:
1536:
1534:
1532:
1529:
1528:
1515:
1512:
1489:
1486:
1474:Turing complete
1463:
1460:
1454:
1439:
1436:
1407:
1403:
1382:
1378:
1368:
1358:
1354:
1314:
1312:
1310:
1307:
1306:
1282:
1281:
1274:
1264:
1262:
1246:
1242:
1229:
1226:
1223:
1222:
1215:
1205:
1203:
1187:
1183:
1170:
1167:
1164:
1163:
1147:
1145:
1136:
1135:
1119:
1117:
1104:
1103:
1086:
1083:
1082:
1069:
1066:
1047:
1044:
1038:
1015:
1012:
983:
979:
952:
948:
927:
923:
916:
914:
897:
894:
893:
863:
859:
838:
834:
824:
810:
808:
806:
803:
802:
789:
786:
748:
744:
737:
735:
718:
715:
714:
659:
654:
652:
649:
648:
635:
632:
601:
597:
579:
575:
568:
566:
549:
546:
545:
490:
476:
474:
472:
469:
468:
451:
448:
423:
255:
254:
250:
246:
166:
156:
155:
84:
80:
71:Stephen Wolfram
66:
17:
12:
11:
5:
5215:
5205:
5204:
5190:
5189:
5184:
5179:
5169:
5159:
5158:External links
5156:
5153:
5152:
5125:(1): 100–111.
5110:(2006-04-01).
5098:
5087:
5076:
5065:
5058:
5025:
5014:
5001:
4950:
4937:
4910:
4909:
4908:
4907:
4888:
4869:
4850:
4831:
4812:
4793:
4774:
4755:
4736:
4717:
4698:
4679:
4660:
4641:
4622:
4603:
4584:
4563:
4560:
4546:
4543:
4542:
4541:
4538:
4531:
4529:
4526:
4519:
4517:
4515:Rule 200 (236)
4514:
4507:
4505:
4499:
4492:
4490:
4487:
4480:
4478:
4475:
4468:
4466:
4464:Rule 170 (240)
4463:
4456:
4454:
4451:
4444:
4442:
4440:Rule 164 (218)
4439:
4432:
4430:
4427:
4420:
4418:
4416:Rule 160 (250)
4415:
4408:
4406:
4404:Rule 156 (198)
4403:
4396:
4394:
4391:
4384:
4382:
4379:
4372:
4370:
4367:
4360:
4358:
4356:Rule 146 (182)
4355:
4348:
4346:
4344:Rule 142 (212)
4343:
4336:
4334:
4331:
4324:
4322:
4319:
4312:
4310:
4307:
4300:
4298:
4295:
4288:
4286:
4284:Rule 132 (222)
4283:
4276:
4274:
4271:
4264:
4262:
4260:Rule 128 (254)
4259:
4252:
4250:
4248:Rule 126 (129)
4247:
4240:
4238:
4236:Rule 122 (161)
4235:
4228:
4226:
4220:
4213:
4211:
4209:Rule 108 (201)
4208:
4201:
4199:
4196:
4189:
4187:
4184:
4177:
4175:
4173:Rule 104 (233)
4172:
4165:
4163:
4160:
4153:
4151:
4145:
4138:
4136:
4133:
4126:
4124:
4121:
4114:
4112:
4109:
4102:
4100:
4097:
4090:
4088:
4085:
4078:
4076:
4073:
4066:
4064:
4061:
4054:
4052:
4049:
4042:
4040:
4037:
4030:
4028:
4025:
4018:
4016:
4013:
4006:
4004:
4001:
3994:
3992:
3989:
3982:
3980:
3977:
3970:
3968:
3965:
3958:
3956:
3953:
3946:
3944:
3941:
3934:
3932:
3929:
3922:
3920:
3917:
3910:
3908:
3905:
3898:
3896:
3893:
3886:
3884:
3881:
3874:
3872:
3869:
3862:
3860:
3857:
3850:
3848:
3845:
3838:
3836:
3833:
3826:
3824:
3821:
3814:
3812:
3809:
3802:
3800:
3798:(86, 135, 149)
3794:
3787:
3785:
3782:
3775:
3773:
3770:
3763:
3761:
3758:
3751:
3749:
3746:
3739:
3737:
3734:
3727:
3725:
3722:
3715:
3713:
3710:
3703:
3701:
3698:
3691:
3689:
3686:
3679:
3677:
3674:
3667:
3665:
3662:
3655:
3653:
3650:
3643:
3641:
3638:
3631:
3629:
3626:
3619:
3617:
3614:
3607:
3605:
3602:
3595:
3593:
3590:
3583:
3581:
3578:
3571:
3569:
3566:
3559:
3557:
3554:
3547:
3545:
3542:
3535:
3533:
3530:
3523:
3521:
3518:
3511:
3509:
3506:
3499:
3497:
3494:
3487:
3485:
3482:
3475:
3465:
3464:
3453:
3450:
3447:
3438:
3435:
3434:
3433:
3430:
3423:
3421:
3418:
3411:
3409:
3406:
3399:
3397:
3394:
3387:
3385:
3382:
3375:
3373:
3370:
3363:
3361:
3358:
3351:
3349:
3346:
3339:
3337:
3334:
3327:
3325:
3322:
3315:
3313:
3310:
3303:
3301:
3298:
3291:
3289:
3286:
3279:
3277:
3274:
3267:
3265:
3262:
3255:
3253:
3250:
3243:
3241:
3238:
3231:
3229:
3226:
3219:
3217:
3214:
3207:
3205:
3202:
3195:
3193:
3190:
3183:
3181:
3178:
3171:
3169:
3166:
3159:
3157:
3154:
3147:
3145:
3142:
3135:
3133:
3130:
3123:
3121:
3118:
3111:
3109:
3106:
3099:
3097:
3094:
3087:
3085:
3082:
3075:
3073:
3070:
3063:
3061:
3058:
3051:
3049:
3046:
3039:
3037:
3034:
3027:
3025:
3022:
3015:
3013:
3010:
3003:
3001:
2998:
2991:
2989:
2986:
2979:
2977:
2974:
2967:
2965:
2962:
2955:
2953:
2950:
2943:
2941:
2938:
2931:
2929:
2926:
2919:
2917:
2914:
2907:
2905:
2902:
2895:
2893:
2890:
2883:
2881:
2878:
2871:
2869:
2866:
2859:
2857:
2854:
2847:
2845:
2842:
2835:
2833:
2830:
2823:
2821:
2818:
2811:
2809:
2806:
2799:
2797:
2794:
2787:
2785:
2782:
2775:
2773:
2770:
2763:
2761:
2758:
2751:
2749:
2746:
2739:
2737:
2734:
2727:
2725:
2722:
2715:
2713:
2710:
2703:
2701:
2698:
2691:
2689:
2686:
2679:
2677:
2674:
2667:
2665:
2662:
2655:
2653:
2650:
2643:
2641:
2638:
2631:
2629:
2626:
2619:
2617:
2614:
2607:
2605:
2602:
2595:
2593:
2590:
2583:
2581:
2578:
2571:
2569:
2566:
2559:
2557:
2554:
2547:
2545:
2542:
2535:
2533:
2530:
2523:
2521:
2518:
2511:
2509:
2506:
2499:
2497:
2494:
2487:
2485:
2482:
2475:
2473:
2470:
2463:
2461:
2458:
2451:
2449:
2446:
2439:
2437:
2434:
2427:
2425:
2422:
2415:
2413:
2410:
2403:
2401:
2398:
2391:
2389:
2386:
2379:
2377:
2374:
2367:
2365:
2362:
2355:
2353:
2350:
2343:
2341:
2338:
2331:
2329:
2326:
2319:
2317:
2314:
2307:
2305:
2302:
2295:
2293:
2290:
2283:
2281:
2278:
2271:
2269:
2266:
2259:
2257:
2254:
2247:
2245:
2242:
2235:
2227:
2224:
2220:
2219:
2207:
2204:
2199:
2195:
2191:
2188:
2185:
2182:
2179:
2176:
2173:
2159:
2158:
2143:
2140:
2137:
2134:
2131:
2128:
2125:
2122:
2119:
2116:
2113:
2108:
2105:
2102:
2099:
2067:
2064:
2060:
2059:
2047:
2044:
2039:
2035:
2031:
2028:
2025:
2022:
2019:
2016:
2013:
1999:
1998:
1983:
1980:
1977:
1974:
1971:
1968:
1965:
1962:
1959:
1956:
1953:
1949:
1918:
1915:
1914:
1913:
1898:
1895:
1892:
1889:
1886:
1883:
1880:
1875:
1871:
1867:
1864:
1861:
1856:
1853:
1850:
1847:
1819:
1816:
1815:
1814:
1799:
1794:
1790:
1786:
1783:
1780:
1777:
1774:
1769:
1765:
1761:
1758:
1755:
1748:
1744:
1740:
1737:
1732:
1728:
1724:
1721:
1716:
1712:
1708:
1705:
1700:
1696:
1692:
1689:
1686:
1683:
1680:
1677:
1649:
1646:
1645:
1644:
1629:
1624:
1620:
1616:
1613:
1610:
1607:
1604:
1599:
1595:
1591:
1588:
1585:
1578:
1574:
1570:
1567:
1562:
1558:
1554:
1551:
1548:
1545:
1542:
1539:
1511:
1508:
1485:
1482:
1456:Main article:
1453:
1450:
1435:
1432:
1431:
1430:
1415:
1410:
1406:
1402:
1399:
1396:
1393:
1390:
1385:
1381:
1377:
1374:
1371:
1366:
1361:
1357:
1353:
1350:
1347:
1344:
1341:
1338:
1335:
1332:
1329:
1326:
1323:
1320:
1317:
1300:
1299:
1285:
1273:
1263:
1261:
1255:
1249:
1245:
1241:
1238:
1235:
1232:
1225:
1224:
1214:
1204:
1202:
1196:
1190:
1186:
1182:
1179:
1176:
1173:
1166:
1165:
1162:
1159:
1156:
1146:
1144:
1141:
1138:
1137:
1134:
1131:
1128:
1118:
1116:
1113:
1110:
1109:
1107:
1102:
1099:
1096:
1093:
1090:
1065:
1062:
1040:Main article:
1037:
1034:
1011:
1008:
1007:
1006:
992:
986:
982:
978:
975:
972:
969:
966:
963:
960:
955:
951:
947:
944:
941:
938:
935:
930:
926:
922:
919:
913:
910:
907:
904:
901:
887:
886:
871:
866:
862:
858:
855:
852:
849:
846:
841:
837:
833:
830:
827:
822:
819:
816:
813:
785:
782:
778:
777:
763:
759:
756:
751:
747:
743:
740:
734:
731:
728:
725:
722:
708:
707:
692:
689:
686:
683:
680:
677:
674:
671:
668:
665:
662:
658:
631:
628:
624:
623:
610:
604:
600:
596:
593:
590:
587:
582:
578:
574:
571:
565:
562:
559:
556:
553:
539:
538:
523:
520:
517:
514:
511:
508:
505:
502:
499:
496:
493:
488:
485:
482:
479:
447:
444:
422:
419:
399:
398:
395:
392:
389:
386:
383:
380:
377:
374:
370:
369:
366:
363:
360:
357:
354:
351:
348:
345:
334:
333:
330:
327:
324:
321:
318:
315:
312:
309:
305:
304:
301:
298:
295:
292:
289:
286:
283:
280:
258:
257:
252:
248:
244:
242:
239:
236:
233:
230:
227:
224:
221:
218:
215:
211:
210:
207:
204:
201:
198:
195:
192:
189:
186:
183:
165:
162:
159:
158:
153:
151:
148:
145:
142:
139:
136:
133:
130:
127:
124:
120:
119:
116:
113:
110:
107:
104:
101:
98:
95:
92:
82:
78:
65:
62:
15:
9:
6:
4:
3:
2:
5214:
5203:
5200:
5199:
5197:
5188:
5185:
5183:
5180:
5177:
5173:
5170:
5168:
5167:
5162:
5161:
5148:
5144:
5140:
5136:
5132:
5128:
5124:
5120:
5113:
5109:
5102:
5096:
5091:
5085:
5080:
5074:
5069:
5061:
5059:0-201-62716-7
5055:
5048:
5047:
5039:
5035:
5029:
5023:
5018:
5011:
5005:
4997:
4993:
4988:
4983:
4978:
4973:
4969:
4965:
4961:
4954:
4947:
4941:
4926:
4922:
4915:
4911:
4903:
4902:
4897:
4894:
4889:
4884:
4883:
4878:
4875:
4870:
4865:
4864:
4859:
4856:
4851:
4846:
4845:
4840:
4837:
4832:
4827:
4826:
4821:
4818:
4813:
4808:
4807:
4802:
4799:
4794:
4789:
4788:
4783:
4780:
4775:
4770:
4769:
4764:
4761:
4756:
4751:
4750:
4745:
4742:
4737:
4732:
4731:
4726:
4723:
4718:
4713:
4712:
4707:
4704:
4699:
4694:
4693:
4688:
4685:
4680:
4675:
4674:
4669:
4666:
4661:
4656:
4655:
4650:
4647:
4642:
4637:
4636:
4631:
4628:
4623:
4618:
4617:
4612:
4609:
4604:
4599:
4598:
4593:
4590:
4585:
4580:
4579:
4574:
4571:
4566:
4565:
4559:
4555:
4551:
4545:Unusual cases
4535:
4530:
4523:
4518:
4511:
4506:
4502:
4496:
4491:
4484:
4479:
4472:
4467:
4460:
4455:
4448:
4443:
4436:
4431:
4424:
4419:
4412:
4407:
4400:
4395:
4388:
4383:
4376:
4371:
4364:
4359:
4352:
4347:
4340:
4335:
4328:
4323:
4316:
4311:
4304:
4299:
4292:
4287:
4280:
4275:
4268:
4263:
4256:
4251:
4244:
4239:
4232:
4227:
4223:
4217:
4212:
4205:
4200:
4193:
4188:
4181:
4176:
4169:
4164:
4161:Rule 94 (133)
4157:
4152:
4148:
4142:
4137:
4130:
4125:
4118:
4113:
4110:Rule 76 (205)
4106:
4101:
4094:
4089:
4086:Rule 73 (109)
4082:
4077:
4074:Rule 72 (237)
4070:
4065:
4058:
4053:
4046:
4041:
4034:
4029:
4022:
4017:
4010:
4005:
4002:Rule 54 (147)
3998:
3993:
3986:
3981:
3978:Rule 50 (179)
3974:
3969:
3962:
3957:
3950:
3945:
3938:
3933:
3930:Rule 43 (113)
3926:
3921:
3914:
3909:
3902:
3897:
3890:
3885:
3878:
3873:
3866:
3861:
3858:Rule 36 (219)
3854:
3849:
3842:
3837:
3830:
3825:
3822:Rule 33 (123)
3818:
3813:
3810:Rule 32 (251)
3806:
3801:
3797:
3791:
3786:
3779:
3774:
3767:
3762:
3755:
3750:
3743:
3738:
3731:
3726:
3719:
3714:
3707:
3702:
3699:Rule 22 (151)
3695:
3690:
3683:
3678:
3675:Rule 18 (183)
3671:
3666:
3659:
3654:
3647:
3642:
3635:
3630:
3623:
3618:
3611:
3606:
3599:
3594:
3587:
3582:
3575:
3570:
3563:
3558:
3551:
3546:
3539:
3534:
3527:
3522:
3515:
3510:
3503:
3498:
3491:
3486:
3479:
3474:
3473:
3472:
3468:
3462:
3458:
3454:
3451:
3448:
3445:
3444:
3443:
3427:
3422:
3415:
3410:
3403:
3398:
3391:
3386:
3379:
3374:
3367:
3362:
3355:
3350:
3343:
3338:
3331:
3326:
3319:
3314:
3307:
3302:
3295:
3290:
3283:
3278:
3271:
3266:
3259:
3254:
3247:
3242:
3235:
3230:
3223:
3218:
3211:
3206:
3199:
3194:
3187:
3182:
3175:
3170:
3163:
3158:
3151:
3146:
3139:
3134:
3127:
3122:
3115:
3110:
3103:
3098:
3091:
3086:
3079:
3074:
3067:
3062:
3055:
3050:
3043:
3038:
3031:
3026:
3019:
3014:
3007:
3002:
2995:
2990:
2983:
2978:
2971:
2966:
2959:
2954:
2947:
2942:
2935:
2930:
2923:
2918:
2911:
2906:
2899:
2894:
2887:
2882:
2875:
2870:
2863:
2858:
2851:
2846:
2839:
2834:
2827:
2822:
2815:
2810:
2803:
2798:
2791:
2786:
2779:
2774:
2767:
2762:
2755:
2750:
2743:
2738:
2731:
2726:
2719:
2714:
2707:
2702:
2695:
2690:
2683:
2678:
2671:
2666:
2659:
2654:
2647:
2642:
2635:
2630:
2623:
2618:
2611:
2606:
2599:
2594:
2587:
2582:
2575:
2570:
2563:
2558:
2551:
2546:
2539:
2534:
2527:
2522:
2515:
2510:
2503:
2498:
2491:
2486:
2479:
2474:
2467:
2462:
2455:
2450:
2443:
2438:
2431:
2426:
2419:
2414:
2407:
2402:
2395:
2390:
2383:
2378:
2371:
2366:
2359:
2354:
2347:
2342:
2335:
2330:
2323:
2318:
2311:
2306:
2299:
2294:
2287:
2282:
2275:
2270:
2263:
2258:
2251:
2246:
2239:
2234:
2233:
2232:
2223:
2205:
2202:
2197:
2193:
2189:
2186:
2183:
2177:
2171:
2164:
2163:
2162:
2138:
2135:
2132:
2129:
2120:
2117:
2114:
2106:
2103:
2100:
2097:
2087:
2086:
2085:
2083:
2079:
2074:
2063:
2045:
2042:
2037:
2033:
2029:
2026:
2023:
2017:
2011:
2004:
2003:
2002:
1978:
1975:
1972:
1969:
1960:
1957:
1954:
1947:
1938:
1937:
1936:
1934:
1930:
1925:
1893:
1890:
1887:
1884:
1873:
1869:
1865:
1862:
1854:
1851:
1848:
1845:
1835:
1834:
1833:
1831:
1826:
1792:
1788:
1784:
1781:
1778:
1767:
1763:
1759:
1756:
1746:
1742:
1738:
1735:
1730:
1726:
1722:
1719:
1714:
1710:
1706:
1703:
1698:
1694:
1690:
1687:
1684:
1681:
1678:
1675:
1665:
1664:
1663:
1661:
1656:
1622:
1618:
1614:
1611:
1608:
1597:
1593:
1589:
1586:
1576:
1572:
1568:
1565:
1560:
1556:
1552:
1549:
1546:
1543:
1540:
1537:
1527:
1526:
1525:
1523:
1518:
1507:
1505:
1501:
1497:
1492:
1481:
1479:
1475:
1471:
1466:
1459:
1449:
1447:
1442:
1408:
1404:
1400:
1397:
1394:
1383:
1379:
1375:
1372:
1359:
1355:
1351:
1348:
1345:
1342:
1339:
1336:
1327:
1324:
1321:
1318:
1305:
1304:
1303:
1271:
1259:
1253:
1247:
1243:
1239:
1236:
1233:
1230:
1212:
1200:
1194:
1188:
1184:
1180:
1177:
1174:
1171:
1160:
1157:
1154:
1142:
1139:
1132:
1129:
1126:
1114:
1111:
1105:
1100:
1094:
1088:
1081:
1080:
1079:
1077:
1072:
1061:
1059:
1055:
1050:
1043:
1033:
1031:
1027:
1023:
1018:
990:
984:
976:
973:
967:
964:
961:
958:
953:
945:
942:
936:
933:
928:
924:
920:
917:
911:
905:
899:
892:
891:
890:
864:
860:
856:
853:
850:
839:
835:
831:
828:
820:
817:
814:
811:
801:
800:
799:
797:
792:
781:
761:
757:
754:
749:
745:
741:
738:
732:
726:
720:
713:
712:
711:
687:
684:
681:
678:
669:
666:
663:
656:
647:
646:
645:
643:
638:
627:
608:
602:
594:
591:
585:
580:
576:
572:
569:
563:
557:
551:
544:
543:
542:
518:
515:
512:
509:
500:
497:
494:
486:
483:
480:
477:
467:
466:
465:
463:
459:
454:
443:
441:
437:
433:
429:
418:
416:
415:automorphisms
411:
408:
404:
396:
393:
390:
387:
384:
381:
378:
375:
372:
371:
367:
364:
361:
358:
355:
352:
349:
346:
343:
342:
339:
331:
328:
325:
322:
319:
316:
313:
310:
307:
306:
302:
299:
296:
293:
290:
287:
284:
281:
278:
277:
274:
272:
267:
265:
240:
237:
234:
231:
228:
225:
222:
219:
216:
213:
212:
208:
205:
202:
199:
196:
193:
190:
187:
184:
181:
180:
177:
174:
172:
171:mirrored rule
149:
146:
143:
140:
137:
134:
131:
128:
125:
122:
121:
117:
114:
111:
108:
105:
102:
99:
96:
93:
90:
89:
86:
76:
72:
61:
59:
55:
50:
46:
42:
38:
29:
21:
5165:
5122:
5118:
5101:
5090:
5079:
5068:
5045:
5028:
5017:
5009:
5004:
4967:
4963:
4953:
4940:
4928:. Retrieved
4924:
4914:
4899:
4880:
4861:
4842:
4823:
4804:
4785:
4766:
4747:
4728:
4709:
4690:
4671:
4652:
4633:
4614:
4595:
4576:
4556:
4552:
4548:
4026:Rule 57 (99)
3870:Rule 37 (91)
3783:Rule 29 (71)
3687:Rule 19 (55)
3663:Rule 15 (85)
3531:Rule 4 (223)
3495:Rule 1 (127)
3483:Rule 0 (255)
3469:
3466:
3440:
2229:
2221:
2160:
2069:
2061:
2000:
1920:
1821:
1651:
1513:
1503:
1499:
1487:
1461:
1437:
1301:
1067:
1045:
1013:
888:
787:
779:
709:
633:
625:
540:
449:
435:
431:
427:
424:
412:
409:
405:
402:
337:
270:
268:
263:
261:
175:
170:
167:
75:Wolfram code
67:
44:
34:
5178:by default)
3543:Rule 5 (95)
264:amphichiral
37:mathematics
4896:"Rule 222"
4877:"Rule 220"
4858:"Rule 190"
4839:"Rule 188"
4820:"Rule 182"
4801:"Rule 158"
4782:"Rule 150"
4763:"Rule 126"
4744:"Rule 110"
4725:"Rule 102"
4562:References
209:P=(L,C,R)
118:P=(L,C,R)
5147:0960-0779
4996:2075-2180
4977:0906.3248
4970:: 31–55.
4901:MathWorld
4882:MathWorld
4863:MathWorld
4844:MathWorld
4825:MathWorld
4806:MathWorld
4787:MathWorld
4768:MathWorld
4749:MathWorld
4730:MathWorld
4711:MathWorld
4706:"Rule 94"
4692:MathWorld
4687:"Rule 90"
4673:MathWorld
4668:"Rule 62"
4654:MathWorld
4649:"Rule 60"
4635:MathWorld
4630:"Rule 54"
4616:MathWorld
4611:"Rule 50"
4597:MathWorld
4592:"Rule 30"
4578:MathWorld
2203:−
2190:⋅
2133:−
2118:−
2043:−
2030:⋅
1973:−
1958:−
1888:−
1866:−
1782:−
1760:−
1736:−
1612:−
1590:−
1566:−
1398:−
1376:−
1349:−
1240:⋅
1181:⋅
974:−
959:−
943:−
934:−
921:⋅
854:−
832:−
755:−
742:⋅
682:−
667:−
592:−
586:−
573:⋅
513:−
81:=01101110
5196:Category
5176:Rule 110
5036:(1994).
5012:p223 ff.
4539:Rule 232
4527:Rule 204
4501:Rule 184
4488:Rule 178
4368:Rule 150
4222:Rule 110
4185:Rule 105
3457:rule 110
2066:Rule 222
1917:Rule 220
1818:Rule 190
1648:Rule 188
1510:Rule 158
1484:Rule 150
1458:Rule 110
1452:Rule 110
1434:Rule 102
1267:if
1208:if
1150:if
1122:if
54:rule 110
5127:Bibcode
4930:June 9,
4925:Fōrmulæ
4147:Rule 90
4122:Rule 77
3990:Rule 51
3796:Rule 30
3711:Rule 23
3431:Rule 99
3419:Rule 98
3407:Rule 97
3395:Rule 96
3383:Rule 95
3371:Rule 94
3359:Rule 93
3347:Rule 92
3335:Rule 91
3323:Rule 90
3311:Rule 89
3299:Rule 88
3287:Rule 87
3275:Rule 86
3263:Rule 85
3251:Rule 84
3239:Rule 83
3227:Rule 82
3215:Rule 81
3203:Rule 80
3191:Rule 79
3179:Rule 78
3167:Rule 77
3155:Rule 76
3143:Rule 75
3131:Rule 74
3119:Rule 73
3107:Rule 72
3095:Rule 71
3083:Rule 70
3071:Rule 69
3059:Rule 68
3047:Rule 67
3035:Rule 66
3023:Rule 65
3011:Rule 64
2999:Rule 63
2987:Rule 62
2975:Rule 61
2963:Rule 60
2951:Rule 59
2939:Rule 58
2927:Rule 57
2915:Rule 56
2903:Rule 55
2891:Rule 54
2879:Rule 53
2867:Rule 52
2855:Rule 51
2843:Rule 50
2831:Rule 49
2819:Rule 48
2807:Rule 47
2795:Rule 46
2783:Rule 45
2771:Rule 44
2759:Rule 43
2747:Rule 42
2735:Rule 41
2723:Rule 40
2711:Rule 39
2699:Rule 38
2687:Rule 37
2675:Rule 36
2663:Rule 35
2651:Rule 34
2639:Rule 33
2627:Rule 32
2615:Rule 31
2603:Rule 30
2591:Rule 29
2579:Rule 28
2567:Rule 27
2555:Rule 26
2543:Rule 25
2531:Rule 24
2519:Rule 23
2507:Rule 22
2495:Rule 21
2483:Rule 20
2471:Rule 19
2459:Rule 18
2447:Rule 17
2435:Rule 16
2423:Rule 15
2411:Rule 14
2399:Rule 13
2387:Rule 12
2375:Rule 11
2363:Rule 10
2076:in the
2073:A083420
1927:in the
1924:A000225
1828:in the
1825:A037576
1658:in the
1655:A118173
1520:in the
1517:A118171
1494:in the
1491:A038184
1468:in the
1465:A117999
1444:in the
1441:A117998
1074:in the
1071:A118101
1064:Rule 94
1052:in the
1049:A038183
1042:Rule 90
1036:Rule 90
1020:in the
1017:A001317
1010:Rule 60
794:in the
791:A118108
784:Rule 54
640:in the
637:A002450
630:Rule 50
456:in the
453:A001045
446:Rule 28
5145:
5056:
4994:
2351:Rule 9
2339:Rule 8
2327:Rule 7
2315:Rule 6
2303:Rule 5
2291:Rule 4
2279:Rule 3
2267:Rule 2
2255:Rule 1
2243:Rule 0
5115:(PDF)
5050:(PDF)
5041:(PDF)
4972:arXiv
4503:(226)
4149:(165)
43:, an
5143:ISSN
5054:ISBN
4992:ISSN
4932:2024
2078:OEIS
1929:OEIS
1830:OEIS
1660:OEIS
1522:OEIS
1496:OEIS
1470:OEIS
1446:OEIS
1076:OEIS
1054:OEIS
1022:OEIS
796:OEIS
642:OEIS
458:OEIS
368:000
365:001
362:010
359:011
356:100
353:101
350:110
347:111
303:111
300:110
297:101
294:100
291:011
288:010
285:001
282:000
251:=124
203:000
200:001
197:010
194:011
191:100
188:101
185:110
182:111
112:000
109:001
106:010
103:011
100:100
97:101
94:110
91:111
39:and
5135:doi
4982:doi
247:+12
243:112
152:110
35:In
5198::
5141:.
5133:.
5123:28
5121:.
5117:.
5043:.
4990:.
4980:.
4966:.
4962:.
4923:.
4898:.
4879:.
4860:.
4841:.
4822:.
4803:.
4784:.
4765:.
4746:.
4727:.
4708:.
4689:.
4670:.
4651:.
4632:.
4613:.
4594:.
4575:.
1785:16
1707:12
1615:16
1553:12
1480:.
1401:16
1352:16
1237:11
1231:10
1032:.
991:15
918:22
857:16
397:1
394:0
391:0
388:1
385:0
382:0
379:0
376:1
332:1
329:0
326:0
323:0
320:1
317:0
314:0
311:1
235:0
232:0
229:1
226:1
223:1
220:1
217:1
214:0
144:0
141:1
138:1
135:1
132:0
129:1
126:1
123:0
5174:(
5149:.
5137::
5129::
5062:.
4998:.
4984::
4974::
4968:1
4948:.
4934:.
4904:.
4885:.
4866:.
4847:.
4828:.
4809:.
4790:.
4771:.
4752:.
4733:.
4714:.
4695:.
4676:.
4657:.
4638:.
4619:.
4600:.
4581:.
3463:.
2218:.
2206:1
2198:t
2194:4
2187:2
2184:=
2181:)
2178:t
2175:(
2172:a
2157:.
2142:)
2139:x
2136:4
2130:1
2127:(
2124:)
2121:x
2115:1
2112:(
2107:x
2104:2
2101:+
2098:1
2058:.
2046:1
2038:t
2034:2
2027:2
2024:=
2021:)
2018:t
2015:(
2012:a
1997:.
1982:)
1979:x
1976:2
1970:1
1967:(
1964:)
1961:x
1955:1
1952:(
1948:1
1912:.
1897:)
1894:x
1891:4
1885:1
1882:(
1879:)
1874:2
1870:x
1863:1
1860:(
1855:x
1852:3
1849:+
1846:1
1813:.
1798:)
1793:4
1789:x
1779:1
1776:(
1773:)
1768:2
1764:x
1757:1
1754:(
1747:5
1743:x
1739:8
1731:4
1727:x
1723:8
1720:+
1715:3
1711:x
1704:+
1699:2
1695:x
1691:4
1688:+
1685:x
1682:3
1679:+
1676:1
1643:.
1628:)
1623:2
1619:x
1609:1
1606:(
1603:)
1598:2
1594:x
1587:1
1584:(
1577:3
1573:x
1569:4
1561:2
1557:x
1550:+
1547:x
1544:7
1541:+
1538:1
1504:x
1502:+
1500:x
1429:.
1414:)
1409:2
1405:x
1395:1
1392:(
1389:)
1384:2
1380:x
1373:1
1370:(
1365:)
1360:4
1356:x
1346:x
1343:5
1340:+
1337:1
1334:(
1331:)
1328:x
1325:2
1322:+
1319:1
1316:(
1298:.
1272:t
1260:,
1254:6
1248:n
1244:4
1234:+
1213:t
1201:,
1195:3
1189:n
1185:4
1178:5
1175:+
1172:1
1161:1
1158:=
1155:t
1143:,
1140:7
1133:0
1130:=
1127:t
1115:,
1112:1
1106:{
1101:=
1098:)
1095:t
1092:(
1089:a
1005:.
985:t
981:)
977:1
971:(
968:3
965:+
962:4
954:t
950:)
946:4
940:(
937:6
929:t
925:4
912:=
909:)
906:t
903:(
900:a
885:.
870:)
865:2
861:x
851:1
848:(
845:)
840:2
836:x
829:1
826:(
821:x
818:7
815:+
812:1
776:.
762:3
758:1
750:t
746:4
739:4
733:=
730:)
727:t
724:(
721:a
706:.
691:)
688:x
685:4
679:1
676:(
673:)
670:x
664:1
661:(
657:1
609:3
603:t
599:)
595:1
589:(
581:t
577:2
570:4
564:=
561:)
558:t
555:(
552:a
537:.
522:)
519:x
516:2
510:1
507:(
504:)
501:x
498:+
495:1
492:(
487:x
484:2
481:+
478:1
436:t
434:(
432:a
428:t
253:d
249:d
245:d
241:N
154:d
150:N
83:2
79:d
52:(
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