A151731 -id:A151731 - cp's OEIS frontend

A151732 First differences of A151731.

0, 2, 4, 8, 6, 12, 8, 14, 16, 24, 14, 20, 24, 20, 16, 36, 32, 44, 44, 36, 36, 48, 40, 48, 46, 60, 70, 44, 44, 64, 60, 60, 76, 72, 64, 64, 84, 94, 102, 68, 84, 88, 116, 92, 120, 126, 126, 104, 128, 104, 96, 84, 116, 130, 114, 92, 112, 112, 132, 140, 160, 162

Offset: 0

N. J. A. Sloane, Jun 15 2009

nonn

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Rémy Sigrist, PARI program for A151732

Cf. A151731.

PARI
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A151733 A151731/2.

Original entry on oeis.org

0, 1, 3, 7, 10, 16, 20, 27, 35, 47, 54, 64, 76, 86, 94, 112, 128, 150, 172, 190, 208, 232, 252, 276, 299, 329, 364, 386, 408, 440, 470, 500, 538, 574, 606, 638, 680, 727, 778, 812, 854, 898, 956, 1002, 1062, 1125, 1188, 1240, 1304, 1356, 1404

Offset: 0

N. J. A. Sloane, Jun 15 2009

nonn

A151725 Number of ON states after n generations of cellular automaton rule described by the rulestring B1/S012345678.

Original entry on oeis.org

0, 1, 9, 13, 33, 37, 57, 77, 121, 125, 145, 165, 209, 237, 297, 373, 465, 469, 489, 509, 553, 581, 641, 717, 809, 837, 897, 981, 1097, 1213, 1409, 1645, 1833, 1837, 1857, 1877, 1921, 1949, 2009, 2085, 2177, 2205, 2265, 2349, 2465, 2581, 2777, 3013

Offset: 0

David Applegate and N. J. A. Sloane, Jun 13 2009

nonn

A cell is turned ON if exactly one of its eight neighbors is ON. An ON cell remains ON forever.
We start with a single ON cell.
Analog of A147562, which is the case when each cell has only four neighbors.
The equivalent Mathematica cellular automaton is obtained with neighborhood weights {{1,1,1},{1,9,1},{1,1,1}}, rule number 261634, and starting configuration {{1}}. [John W. Layman, Sep 11 2009]
Observation: Visual pattern similar to the toothpick structure (see A139250). [Omar E. Pol, Dec 14 2009]

David Applegate, Table of n, a(n) for n = 0..1000
David Applegate, The movie version
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
N. H. Packard and S. Wolfram, Two-Dimensional Cellular Automata, Journal of Statistical Physics, 38 (1985), 901-946. (See page 920, Figure 7e). Alternative copy
Rémy Sigrist, Illustration of the structure at stage 513
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139250, A151726, A147562, A147582, A151723, A151735, A151747, A151728.

See A151731, A151732, A151733, A151734 for the same CA except that two neighbors must be ON for a cell to turn ON.

RasterGraphics[state_?MatrixQ, colors_Integer : 2, opts___] := Graphics[Raster[ Reverse[1 - state/(colors - 1)]], AspectRatio -> (AspectRatio /. {opts} /. AspectRatio -> Automatic), Frame -> True, FrameTicks -> None, GridLines -> None]; wt = {{1,1,1}, {1,9,1}, {1,1,1}}; rule= 261634; init={{1}}; Show[GraphicsArray[Map[RasterGraphics, CellularAutomaton[{rule, {2, wt}, {1, 1}}, {init, 0}, 9, -10]]]];nx = 100; ca = CellularAutomaton[{rule, {2, wt}, {1, 1}}, {init, 0}, nx - 1, -nx]; a = Table[Total[ca[[i]], 2], {i, 1, nx}] (* John W. Layman, Sep 11 2009 *)
A151725[0] = 0; A151725[n_] := Total[CellularAutomaton[{174766, {2, {{2, 2, 2}, {2, 1, 2}, {2, 2, 2}}}, {1, 1}}, {{{1}}, 0}, {{{n - 1}}}], 2]; Array[A151725, 48, 0] (* JungHwan Min, Sep 01 2016 *)
A151725L[n_] := Prepend[Total[#, 2] & /@ CellularAutomaton[{174766, {2, {{2, 2, 2}, {2, 1, 2}, {2, 2, 2}}}, {1, 1}}, {{{1}}, 0}, n - 1], 0]; A151725L[47] (* JungHwan Min, Sep 01 2016 *)

For a recurrence see the Applegate-Pol-Sloane paper.

Definition clarified by SiYang Hu, May 10 2025

A338097 Number of ON cells after n generations of cellular automaton based on a hexagonal grid and starting with two adjacent ON cells where an OFF cell is turned ON if exactly two of its six neighbors are ON.

Original entry on oeis.org

0, 2, 4, 8, 12, 18, 22, 28, 32, 36, 44, 56, 76, 86, 100, 116, 130, 144, 164, 178, 198, 218, 242, 260, 282, 306, 350, 386, 414, 446, 460, 484, 516, 556, 586, 638, 688, 736, 770, 800, 828, 872, 914, 964, 1012, 1068, 1122, 1162, 1204, 1252, 1300, 1360, 1408, 1462

Offset: 0

Rémy Sigrist, Oct 10 2020

nonn

An ON cell remains ON forever.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Rémy Sigrist, Illustration of initial terms
Rémy Sigrist, Illustration of the structure at generation 512
Rémy Sigrist, PARI program for A338097
Index entries for sequences related to cellular automata

Cf. A151731 (square grid variant), A338098.

PARI
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A249668 Population of the n-th generation of the pattern 101//010 in the Life Without Death cellular automaton.

Original entry on oeis.org

3, 4, 7, 10, 15, 20, 25, 30, 38, 47, 55, 63, 69, 79, 91, 96, 105, 112, 124, 134, 139, 144, 152, 157, 163, 168, 176, 183, 187, 195, 205, 217, 223, 229, 239, 247, 259, 273, 285, 289, 295, 303, 311, 323, 334, 339, 343, 351, 363, 375, 383, 389, 397, 405, 413, 423

Offset: 0

Eric M. Schmidt, Nov 03 2014

nonn

Each generation, a cell turns on if it has exactly three neighbors that are on. Cells never turn off.

This pattern grows indefinitely. No other connected 3-celled pattern does so.

			Generation 0:
101
010
Generation 1:
111
010
Generation 2:
010
111
111
Generation 3:
111
111
111
010
Generation 4:
00100
01110
11111
01110
01110

Eric M. Schmidt, Table of n, a(n) for n = 0..2500
LifeWiki, Life without death
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1,-1). (Initial terms do not obey recurrence; see formula section.)

Cf. A097981, A151725, A151731.

For n >= 2108, a(n+6) = a(n) + 260. - Eric M. Schmidt, Nov 04 2014

For n >= 2115, a(n) = a(n-1) + a(n-6) - a(n-7). - Eric M. Schmidt, Nov 05 2014

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A151732 First differences of A151731.

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A151733 A151731/2.

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A151725 Number of ON states after n generations of cellular automaton rule described by the rulestring B1/S012345678.

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A338097 Number of ON cells after n generations of cellular automaton based on a hexagonal grid and starting with two adjacent ON cells where an OFF cell is turned ON if exactly two of its six neighbors are ON.

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A249668 Population of the n-th generation of the pattern 101//010 in the Life Without Death cellular automaton.

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