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The following are some of the '''integer sequences''' from the [[Game of Life]]. | {{stub}} | ||
The following are some<!--See talk page--> of the '''integer sequences''' from the [[Game of Life]]. | |||
==Still lifes== | ==Still lifes== | ||
* number of n-[[cell]] [[strict still life]]s ({{OEIS|A019473}}-[[Robert Munafo]]): | * number of n-[[cell]] [[strict still life]]s ({{OEIS|A019473}} - [[Robert Munafo]]): | ||
: 0, 0, 0, 2, 1, 5, 4, 9, 10, 25, 46, 121, 240, 619, 1353, 3286, 7773, 19044, 45759, 112243, 273188, 672172, 1646147, 4051732, 9971377, 24619307, 60823008, 150613157, 373188952, 926068847, 2299616637, 5716948683, 14223867298, 35422864104, ... | |||
* number of n-[[cell]] [[pseudo still life]]s ({{OEIS|A056613}}-N. J. A. Sloane): | * number of n-[[cell]] [[pseudo still life]]s ({{OEIS|A056613}} - N. J. A. Sloane): | ||
: 0, 0, 0, 0, 0, 0, 0, 1, 1, 7, 16, 55, 110, 279, 620, 1645, 4067, 10843, 27250, 70637, 179011, 462086, 1184882, 3069135, 7906676, 20463274, 52816265, 136655095, 353198379, 914075620, 2364815358, 6123084116, 15851861075, 4105817368, ... | |||
* number of n-[[cell]] [[quasi still life]]s ({{OEIS|A330283}}-[[Nathaniel Johnston]]): | * number of n-[[cell]] [[quasi still life]]s ({{OEIS|A330283}} - [[Nathaniel Johnston]]): | ||
: 0, 0, 0, 0, 0, 0, 0, 6, 13, 57, 141, 465, 1224, 3956, 11599, 36538, 107415, 327250, 972040, 2957488, 8879327, 26943317, ... | |||
* maximum population of {{times|2|n}} [[still life]]s ({{OEIS|A273308}}-[[Nathaniel Johnston]]): | * maximum population of {{times|2|n}} [[still life]]s ({{OEIS|A273308}} - [[Nathaniel Johnston]]): | ||
: 0, 4, 4, 6, 8, 8, 10, 12, 12, 14, 16, 16, 18, 20, 20, 22, 24, 24, 26, 28, 28, 30, 32, 32, 34, 36, 36, 38, 40, 40, 42, 44, 44, 46, 48, 48, 50, 52, 52, 54, 56, 56, 58, 60, 60, 62, 64, 64, 66, 68, 68, 70, 72, 72, 74, 76, 76, 78, 80, 80, 82, 84, 84, 86, 88, 88, ... | |||
* maximum population of {{times|n|n}} [[still life]]s ({{OEIS|A055397}}-[[Stephen Silver]]): | * maximum population of {{times|n|n}} [[still life]]s ({{OEIS|A055397}} - [[Stephen Silver]]):<ref group="n">See [[User:Apple Bottom/Incubator/Dense stable pattern|Dense stable pattern]]</ref> | ||
: 0, 4, 6, 8, 16, 18, 28, 36, 43, 54, 64, 76, 90, 104, 119, 136, 152, 171, 190, 210, 232, 253, 276, 301, 326, 352, 379, 407, 437, 467, 497, 531, 563, 598, 633, 668, 706, 744, 782, 824, 864, 907, 949, 993, 1039, 1085, 1132, 1181, 1229, 1280, 1331, 1382, 1436, ... | |||
* number of distinct [[lakes]] with 8n [[cell]]s ({{OEIS|A156228}}-[[Nathaniel Johnston]]): | * number of distinct [[lakes]] with 8n [[cell]]s ({{OEIS|A156228}} - [[Nathaniel Johnston]]): | ||
: 1, 0, 1, 1, 4, 7, 31, 98, 446, 1894, 9049, 43151, ... | |||
==Sawtooths== | ==Sawtooths== | ||
* the n<sup>th</sup> [[generation]]s of [[Sawtooth 177]] that have the minimum [[population]] of 177: | * the n<sup>th</sup> [[generation]]s of [[Sawtooth 177]] that have the minimum [[population]] of 177: | ||
: 15, 6975, 849135, 102750495, 12432815055, 1504370626815, 182028845849775, ..., 58(121<sup>n</sup>-1)+15, ... | |||
* the n<sup>th</sup> [[generation]]s of [[Sawtooth 181]] that have the minimum [[population]] of 181: | * the n<sup>th</sup> [[generation]]s of [[Sawtooth 181]] that have the minimum [[population]] of 181: | ||
: 0, 6960, 849120, 102750480, 12432815040, 1504370626800, 182028845849760, ..., 58(121<sup>n</sup>-1), ... | |||
* the n<sup>th</sup> [[generation]]s of [[Sawtooth 201]] that have the minimum [[population]] of 201 ({{OEIS|A257319}}-[[Adam P. Goucher]]): | * the n<sup>th</sup> [[generation]]s of [[Sawtooth 201]] that have the minimum [[population]] of 201 ({{OEIS|A257319}} - [[Adam P. Goucher]]): | ||
: 0, 1840, 88320, 4152880, 195187200, 9173800240, 431168613120, 20264924818480, 952451466470400, 44765218924110640, 2103965289433201920, 98886368603360492080, 4647659324357943129600, 218439988244823327093040, 10266679447506696373374720, ..., 40(47<sup>n</sup>-1), ... | |||
* the n<sup>th</sup> [[generation]]s of [[Sawtooth 362]] that have the minimum [[population]] of 362: | * the n<sup>th</sup> [[generation]]s of [[Sawtooth 362]] that have the minimum [[population]] of 362: | ||
: 1988, 13508, 59588, 243908, 981188, 3930308, 15726788, ..., 960(4<sup>n</sup>)-1852, ... | |||
* the n<sup>th</sup> [[generation]]s of [[Sawtooth 562]] that return to [[population]] 562: | * the n<sup>th</sup> [[generation]]s of [[Sawtooth 562]] that return to [[population]] 562: | ||
: 420, 2940, 18060, 108780, 653100, 3919020, 23514540, 141087660, ..., 14(6<sup>n</sup>)-84, ... | |||
* the n<sup>th</sup> [[generation]]s of [[Sawtooth 1846]] that return to [[population]] 1846: | * the n<sup>th</sup> [[generation]]s of [[Sawtooth 1846]] that return to [[population]] 1846: | ||
: 2625, 92625, 2342625, 58592625, 1464842625, 36621092625, 915527342625, 22888183592625, 572204589842625, 14305114746092600, 357627868652343000, 8940696716308590000, 223517417907715000000, 5587935447692870000000, 6(25<sup>n</sup>)-1125, ... | |||
* the n<sup>th</sup> [[generation]]s of [[Hacksaw]] that return to [[population]] 977: | * the n<sup>th</sup> [[generation]]s of [[Hacksaw]] that return to [[population]] 977: | ||
: 976, 10216, 93376, 841816, 7577776, 68201416, 613814176, 5524329016, 49718962576, 447470664616, ..., 1155(9<sup>n</sup>)-179, ... | |||
==Other sequences== | ==Other sequences== | ||
* number of n-[[cell]] [[ | * number of n-[[cell]] [[polyomino]]es ({{OEIS|A000105}} - N. J. A. Sloane): | ||
: 1, 1, 1, 2, 5, 12, 35, 108, 369, 1285, 4655, 17073, 63600, 238591, 901971, 3426576, 13079255, 50107909, 192622052, 742624232, 2870671950, 11123060678, 43191857688, 168047007728, 654999700403, 2557227044764, 9999088822075, 39153010938487, 153511100594603, ... | |||
* number of n-[[cell]] [[polyplet]]s ({{OEIS|A030222}}-Matthew Cook): | * number of n-[[cell]] [[polyplet]]s ({{OEIS|A030222}} - [[Matthew Cook]]): | ||
: 1, 2, 5, 22, 94, 524, 3031, 18770, 118133, 758381, 4915652, 32149296, 211637205, 1401194463, 9321454604, 62272330564, 417546684096, ... | |||
* number of n-[[cell]] [[period]]-2 [[oscillator]]s ({{OEIS|A056614}}-N. J. A. Sloane): | * number of n-[[cell]] [[period]]-2 [[oscillator]]s ({{OEIS|A056614}} - N. J. A. Sloane): | ||
: 0, 0, 1, 0, 0, 3, 0, 1, 1, 1, 1, 6, 3, 20, 29, 98, 199, 484, 1083, 2722, 6596, ... | |||
* [[period]] of the [[pattern]] that results from the [[evolution]] of a [[one-cell-thick pattern]] of length n ({{OEIS|A061342}}-Alex Fink)): | * [[period]] of the [[pattern]] that results from the [[evolution]] of a [[one-cell-thick pattern]] of length n ({{OEIS|A061342}} - Alex Fink)):<ref>{{LinkForumThread|format=ref|p=160142|title=Period of stationary component of a row of n cells becomes}}</ref> | ||
: 1, 1, 2, 1, 2, 1, 1, 1, 2, 15, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, ... | |||
* number of [[generation]]s for a [[one-cell-thick pattern]] of length n to stabilize ({{OEIS|A152389}}-N. J. A. Sloane): | * number of [[generation]]s for a [[one-cell-thick pattern]] of length n to stabilize ({{OEIS|A152389}} - N. J. A. Sloane): | ||
: 0, 1, 1, 0, 2, 6, 12, 14, 48, 20, 2, 15, 15, 24, 28, 40, 32, 24, 20, 25, 20, 19, 35, 30, 28, 93, 24, 28, 33, 36, 103, 148, 60, 580, 42, 57, 91, 106, 262, 276, 49, 209, 57, 52, 56, 97, 54, 168, 194, 811, 103, 52, 52, 83, 57, 79, 246, 416, 62, 62, 312, 115, 116, ... | |||
* [[population]] of [[lumps of muck]] sequence of [[pattern]]s evolving from the [[grandfather]] of [[stairstep hexomino]] to a [[blockade]] ({{OEIS|A179415}}-Antti Karttunen): | * [[population]] of [[lumps of muck]] sequence of [[pattern]]s evolving from the [[grandfather]] of [[stairstep hexomino]] to a [[blockade]] ({{OEIS|A179415}} - Antti Karttunen): | ||
: 6, 6, 6, 8, 10, 12, 16, 18, 20, 26, 24, 28, 30, 22, 32, 28, 32, 36, 48, 42, 56, 34, 26, 28, 40, 38, 50, 48, 46, 64, 48, 46, 48, 46, 48, 56, 52, 66, 62, 66, 68, 86, 60, 70, 64, 72, 50, 50, 50, 40, 42, 46, 48, 36, 38, 36, 42, 48, 46, 44, 34, 30, 26, 22, 20, 16, 16, 16, 16, 16, ... | |||
* number of [[parent]]s of successive approximations used in a greedy approach to creating a [[Garden of Eden]] ({{OEIS|A196447}}-[[Nicolay Beluchenko]]): | * number of [[parent]]s of successive approximations used in a greedy approach to creating a [[Garden of Eden]] ({{OEIS|A196447}} - [[Nicolay Beluchenko]]): | ||
: 140, 417, 1164, 1005, 3141, 2835, 8797, 7918, 7268, 23415, 21576, 20648, 65342, 62390, 60038, 59165, 177559, 158105, 144487, 136744, 398009, 345711, 317176, 293203, 256688, 822470, 760976, 731808, 714462, 650945, 2087659, 1914317, 1818736, 1811165, 1670837, ... | |||
* numbers of [[cell]]s in which the minimum quantity of [[parent]]s is reached at an OFF state in a [[Garden of Eden]] created by a greedy approach({{OEIS|A197734}}-[[Nicolay Beluchenko]]): | * numbers of [[cell]]s in which the minimum quantity of [[parent]]s is reached at an OFF state in a [[Garden of Eden]] created by a greedy approach ({{OEIS|A197734}} - [[Nicolay Beluchenko]]): | ||
: 40, 46, 53, 61, 68, 72, 79, 85, 98, 113, 117, 121, 123, 130, 137, 146, 151, 155, 159, 164, 174, 178, 186, 190, 195, 200, 206, 212, 217, 218, 222, 225, 234, 235, 239, 243, 247, 253, 254, 256, 263, ... | |||
* number of [[cell]]s in the [[Moore neighbourhood]] of a single cell in an n-dimensional [[cellular automaton]] ({{OEIS|A024023}} - N. J. A. Sloane): | |||
: 0, 2, 8, 26, 80, 242, 728, 2186, 6560, 19682, 59048, 177146, 531440, 1594322, 4782968, 14348906, 43046720, 129140162, 387420488, 1162261466, 3486784400, 10460353202, 31381059608, 94143178826, 282429536480, 847288609442, 2541865828328, 7625597484986, 22876792454960, ..., 3<sup>n</sup>-1, ... | |||
* number of [[cell]]s in the [[Moore neighbourhood]] of range n ({{OEIS|A033996}} - N. J. A. Sloane): | |||
: 0, 8, 24, 48, 80, 120, 168, 224, 288, 360, 440, 528, 624, 728, 840, 960, 1088, 1224, 1368, 1520, ..., (2n+1)<sup>2</sup>-1, ... | |||
* number of [[cell]]s in the [[von Neumann neighbourhood]] of a single cell in an n-dimensional [[cellular automaton]] ({{OEIS|A024023}} - N. J. A. Sloane): | |||
: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, ..., 2n, ... | |||
* number of [[spaceship]]s between successive gaps in the [[LWSS]] [[stream]] generated by the [[ruler]] pattern ({{OEIS|A001511}} - N. J. A. Sloane): | |||
: 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 6, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 7, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 6, 1, 2, 1, 3, 1, 2, 1, 4, 1, ... | |||
* integers generated by the [[Collatz 5n+1 simulator]] ({{OEIS|A259193}} starting at 5<sup>th</sup> term - Alonso del Arte): | |||
: 7, 36, 18, 9, 46, 23, 116, 58, 29, 146, 73, 366, 183, 916, 458, 229, 1146, 573, 2866, 1433, 7166, 3583, 17916, 8958, 4479, 22396, 11198, 5599, 27996, 13998, 6999, 34996, 17498, 8749, 43746, 21873, 109366, 54683, 273416, 136708, 68354, 34177, 170886, 85443, ...? (fate unknown) | |||
==[[Isotropic non-totalistic]] neighbourhood transitions== | |||
* number of transitions in ''n''-state von Neumann rules are the doubly triangular numbers ({{OEIS|A002817}} - N. J. A. Sloane) | |||
: 0, 1, 6, 21, 55, 120, 231, 406, 666, 1035, 1540, 2211, 3081, 4186, 5565, 7260, 9316, 11781, 14706, 18145, 22155, 26796, 32131, 38226, 45150, 52975, 61776, 71631, 82621, 94830, 108345, 123256, 139656, 157641, 177310, 198765, 222111, 247456, 274911, 304590, ... | |||
* transitions in ''n''-dimensional [[Margolus neighbourhood]]s are equivalent to irreducible binary functions of ''n'' variables (reflections of the transition are NOTs of a variable, rotations are combinations of reflections and permutations of axes) ({{OEIS|A000616}} - N. J. A. Sloane) | |||
: 1, 2, 3, 6, 22, 402, 1228158, 400507806843728, 527471432057653004017274030725792, 11218076601767519586965281984173341005925142853855481024470471657123840, ... | |||
==Notes== | |||
<references group="n"/> | |||
==See also== | |||
* [[Pólya enumeration theorem]] (contains more sequences of neighbourhood transitions for specific dimensions, with equations with respect to width, and explains their derivation) | |||
==References== | |||
<references> | |||
</references> | |||
== | ==Links== | ||
* [https://oeis.org/ The On-Line Encyclopedia of Integer | * [https://oeis.org/ The On-Line Encyclopedia of Integer Sequences] | ||
* {{LinkForumThread|f=12|t=3379|title=What are your favourite OEIS sequences?}} | |||
* {{LinkForumThread|f=12|t=5975|title=OEIS sequence request thread}} | |||
[[Category: | [[Category:Mathematical miscellany]] | ||
Latest revision as of 09:04, 6 April 2023
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The following are some of the integer sequences from the Game of Life.
Still lifes
- number of n-cell strict still lifes (
A019473 - Robert Munafo):
- 0, 0, 0, 2, 1, 5, 4, 9, 10, 25, 46, 121, 240, 619, 1353, 3286, 7773, 19044, 45759, 112243, 273188, 672172, 1646147, 4051732, 9971377, 24619307, 60823008, 150613157, 373188952, 926068847, 2299616637, 5716948683, 14223867298, 35422864104, ...
- number of n-cell pseudo still lifes ( A056613 - N. J. A. Sloane):
- 0, 0, 0, 0, 0, 0, 0, 1, 1, 7, 16, 55, 110, 279, 620, 1645, 4067, 10843, 27250, 70637, 179011, 462086, 1184882, 3069135, 7906676, 20463274, 52816265, 136655095, 353198379, 914075620, 2364815358, 6123084116, 15851861075, 4105817368, ...
- number of n-cell quasi still lifes ( A330283 - Nathaniel Johnston):
- 0, 0, 0, 0, 0, 0, 0, 6, 13, 57, 141, 465, 1224, 3956, 11599, 36538, 107415, 327250, 972040, 2957488, 8879327, 26943317, ...
- maximum population of 2 × n still lifes ( A273308 - Nathaniel Johnston):
- 0, 4, 4, 6, 8, 8, 10, 12, 12, 14, 16, 16, 18, 20, 20, 22, 24, 24, 26, 28, 28, 30, 32, 32, 34, 36, 36, 38, 40, 40, 42, 44, 44, 46, 48, 48, 50, 52, 52, 54, 56, 56, 58, 60, 60, 62, 64, 64, 66, 68, 68, 70, 72, 72, 74, 76, 76, 78, 80, 80, 82, 84, 84, 86, 88, 88, ...
- maximum population of n × n still lifes ( A055397 - Stephen Silver):[n 1]
- 0, 4, 6, 8, 16, 18, 28, 36, 43, 54, 64, 76, 90, 104, 119, 136, 152, 171, 190, 210, 232, 253, 276, 301, 326, 352, 379, 407, 437, 467, 497, 531, 563, 598, 633, 668, 706, 744, 782, 824, 864, 907, 949, 993, 1039, 1085, 1132, 1181, 1229, 1280, 1331, 1382, 1436, ...
- number of distinct lakes with 8n cells ( A156228 - Nathaniel Johnston):
- 1, 0, 1, 1, 4, 7, 31, 98, 446, 1894, 9049, 43151, ...
Sawtooths
- the nth generations of Sawtooth 177 that have the minimum population of 177:
- 15, 6975, 849135, 102750495, 12432815055, 1504370626815, 182028845849775, ..., 58(121n-1)+15, ...
- the nth generations of Sawtooth 181 that have the minimum population of 181:
- 0, 6960, 849120, 102750480, 12432815040, 1504370626800, 182028845849760, ..., 58(121n-1), ...
- the nth generations of Sawtooth 201 that have the minimum population of 201 ( A257319 - Adam P. Goucher):
- 0, 1840, 88320, 4152880, 195187200, 9173800240, 431168613120, 20264924818480, 952451466470400, 44765218924110640, 2103965289433201920, 98886368603360492080, 4647659324357943129600, 218439988244823327093040, 10266679447506696373374720, ..., 40(47n-1), ...
- the nth generations of Sawtooth 362 that have the minimum population of 362:
- 1988, 13508, 59588, 243908, 981188, 3930308, 15726788, ..., 960(4n)-1852, ...
- the nth generations of Sawtooth 562 that return to population 562:
- 420, 2940, 18060, 108780, 653100, 3919020, 23514540, 141087660, ..., 14(6n)-84, ...
- the nth generations of Sawtooth 1846 that return to population 1846:
- 2625, 92625, 2342625, 58592625, 1464842625, 36621092625, 915527342625, 22888183592625, 572204589842625, 14305114746092600, 357627868652343000, 8940696716308590000, 223517417907715000000, 5587935447692870000000, 6(25n)-1125, ...
- the nth generations of Hacksaw that return to population 977:
- 976, 10216, 93376, 841816, 7577776, 68201416, 613814176, 5524329016, 49718962576, 447470664616, ..., 1155(9n)-179, ...
Other sequences
- number of n-cell polyominoes ( A000105 - N. J. A. Sloane):
- 1, 1, 1, 2, 5, 12, 35, 108, 369, 1285, 4655, 17073, 63600, 238591, 901971, 3426576, 13079255, 50107909, 192622052, 742624232, 2870671950, 11123060678, 43191857688, 168047007728, 654999700403, 2557227044764, 9999088822075, 39153010938487, 153511100594603, ...
- number of n-cell polyplets ( A030222 - Matthew Cook):
- 1, 2, 5, 22, 94, 524, 3031, 18770, 118133, 758381, 4915652, 32149296, 211637205, 1401194463, 9321454604, 62272330564, 417546684096, ...
- number of n-cell period-2 oscillators ( A056614 - N. J. A. Sloane):
- 0, 0, 1, 0, 0, 3, 0, 1, 1, 1, 1, 6, 3, 20, 29, 98, 199, 484, 1083, 2722, 6596, ...
- period of the pattern that results from the evolution of a one-cell-thick pattern of length n ( A061342 - Alex Fink)):[1]
- 1, 1, 2, 1, 2, 1, 1, 1, 2, 15, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, ...
- number of generations for a one-cell-thick pattern of length n to stabilize ( A152389 - N. J. A. Sloane):
- 0, 1, 1, 0, 2, 6, 12, 14, 48, 20, 2, 15, 15, 24, 28, 40, 32, 24, 20, 25, 20, 19, 35, 30, 28, 93, 24, 28, 33, 36, 103, 148, 60, 580, 42, 57, 91, 106, 262, 276, 49, 209, 57, 52, 56, 97, 54, 168, 194, 811, 103, 52, 52, 83, 57, 79, 246, 416, 62, 62, 312, 115, 116, ...
- population of lumps of muck sequence of patterns evolving from the grandfather of stairstep hexomino to a blockade ( A179415 - Antti Karttunen):
- 6, 6, 6, 8, 10, 12, 16, 18, 20, 26, 24, 28, 30, 22, 32, 28, 32, 36, 48, 42, 56, 34, 26, 28, 40, 38, 50, 48, 46, 64, 48, 46, 48, 46, 48, 56, 52, 66, 62, 66, 68, 86, 60, 70, 64, 72, 50, 50, 50, 40, 42, 46, 48, 36, 38, 36, 42, 48, 46, 44, 34, 30, 26, 22, 20, 16, 16, 16, 16, 16, ...
- number of parents of successive approximations used in a greedy approach to creating a Garden of Eden ( A196447 - Nicolay Beluchenko):
- 140, 417, 1164, 1005, 3141, 2835, 8797, 7918, 7268, 23415, 21576, 20648, 65342, 62390, 60038, 59165, 177559, 158105, 144487, 136744, 398009, 345711, 317176, 293203, 256688, 822470, 760976, 731808, 714462, 650945, 2087659, 1914317, 1818736, 1811165, 1670837, ...
- numbers of cells in which the minimum quantity of parents is reached at an OFF state in a Garden of Eden created by a greedy approach ( A197734 - Nicolay Beluchenko):
- 40, 46, 53, 61, 68, 72, 79, 85, 98, 113, 117, 121, 123, 130, 137, 146, 151, 155, 159, 164, 174, 178, 186, 190, 195, 200, 206, 212, 217, 218, 222, 225, 234, 235, 239, 243, 247, 253, 254, 256, 263, ...
- number of cells in the Moore neighbourhood of a single cell in an n-dimensional cellular automaton ( A024023 - N. J. A. Sloane):
- 0, 2, 8, 26, 80, 242, 728, 2186, 6560, 19682, 59048, 177146, 531440, 1594322, 4782968, 14348906, 43046720, 129140162, 387420488, 1162261466, 3486784400, 10460353202, 31381059608, 94143178826, 282429536480, 847288609442, 2541865828328, 7625597484986, 22876792454960, ..., 3n-1, ...
- number of cells in the Moore neighbourhood of range n ( A033996 - N. J. A. Sloane):
- 0, 8, 24, 48, 80, 120, 168, 224, 288, 360, 440, 528, 624, 728, 840, 960, 1088, 1224, 1368, 1520, ..., (2n+1)2-1, ...
- number of cells in the von Neumann neighbourhood of a single cell in an n-dimensional cellular automaton ( A024023 - N. J. A. Sloane):
- 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, ..., 2n, ...
- number of spaceships between successive gaps in the LWSS stream generated by the ruler pattern ( A001511 - N. J. A. Sloane):
- 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 6, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 7, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 6, 1, 2, 1, 3, 1, 2, 1, 4, 1, ...
- integers generated by the Collatz 5n+1 simulator ( A259193 starting at 5th term - Alonso del Arte):
- 7, 36, 18, 9, 46, 23, 116, 58, 29, 146, 73, 366, 183, 916, 458, 229, 1146, 573, 2866, 1433, 7166, 3583, 17916, 8958, 4479, 22396, 11198, 5599, 27996, 13998, 6999, 34996, 17498, 8749, 43746, 21873, 109366, 54683, 273416, 136708, 68354, 34177, 170886, 85443, ...? (fate unknown)
Isotropic non-totalistic neighbourhood transitions
- number of transitions in n-state von Neumann rules are the doubly triangular numbers ( A002817 - N. J. A. Sloane)
- 0, 1, 6, 21, 55, 120, 231, 406, 666, 1035, 1540, 2211, 3081, 4186, 5565, 7260, 9316, 11781, 14706, 18145, 22155, 26796, 32131, 38226, 45150, 52975, 61776, 71631, 82621, 94830, 108345, 123256, 139656, 157641, 177310, 198765, 222111, 247456, 274911, 304590, ...
- transitions in n-dimensional Margolus neighbourhoods are equivalent to irreducible binary functions of n variables (reflections of the transition are NOTs of a variable, rotations are combinations of reflections and permutations of axes) ( A000616 - N. J. A. Sloane)
- 1, 2, 3, 6, 22, 402, 1228158, 400507806843728, 527471432057653004017274030725792, 11218076601767519586965281984173341005925142853855481024470471657123840, ...
Notes
See also
- Pólya enumeration theorem (contains more sequences of neighbourhood transitions for specific dimensions, with equations with respect to width, and explains their derivation)
References
- ↑ Period of stationary component of a row of n cells becomes (discussion thread) at the ConwayLife.com forums
Links
- The On-Line Encyclopedia of Integer Sequences
- What are your favourite OEIS sequences? (discussion thread) at the ConwayLife.com forums
- OEIS sequence request thread (discussion thread) at the ConwayLife.com forums