Integer sequences

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):

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 n^th generations of Sawtooth 177 that have the minimum population of 177:

15, 6975, 849135, 102750495, 12432815055, 1504370626815, 182028845849775, ..., 58(121ⁿ-1)+15, ...

• the n^th generations of Sawtooth 181 that have the minimum population of 181:

0, 6960, 849120, 102750480, 12432815040, 1504370626800, 182028845849760, ..., 58(121ⁿ-1), ...

• the n^th 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 n^th generations of Sawtooth 362 that have the minimum population of 362:

1988, 13508, 59588, 243908, 981188, 3930308, 15726788, ..., 960(4ⁿ)-1852, ...

• the n^th generations of Sawtooth 562 that return to population 562:

420, 2940, 18060, 108780, 653100, 3919020, 23514540, 141087660, ..., 14(6ⁿ)-84, ...

• the n^th 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 n^th 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, 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 5^th 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, ...

See also

• Pólya enumeration theorem (contains more sequences of neighbourhood transitions for specific dimensions, with equations with respect to width, and explains their derivation)

Links

• The On-Line Encyclopedia of Integer Sequences