A179409 -id:A179409 - cp's OEIS frontend

A179410 The number of columns with alive cells in the n-th generation of cyclic sequence of patterns given in A179409, played in Conway's Game of Life on the 8x8 toroidal grid.

4, 5, 3, 3, 5, 5, 7, 6, 7, 8, 7, 7, 7, 4, 7, 5, 6, 6, 5, 5, 5, 5, 7, 3, 4, 5, 3, 3, 5, 5, 7, 6, 7, 8, 7, 7, 7, 4, 7, 5, 6, 6, 5, 5, 5, 5, 7, 3, 4, 5, 3, 3, 5, 5, 7, 6, 7, 8, 7, 7, 7, 4, 7, 5, 6, 6, 5, 5, 5, 5, 7, 3

Offset: 0

Antti Karttunen, Jul 27 2010

nonn

The mean value of terms in the whole period of 24 is 5.5

			See illustrations in A179409. In the first four patterns there are alive cells in 4, 5, 3 and 3 columns, thus a(0)=4, a(1)=5, a(2)=3 and a(3)=3.

A. Karttunen, J-program for computing the terms of this sequence

A179409 & A179411.

A179411 The number of rows with alive cells in the n-th generation of cyclic sequence of patterns given in A179409, played in Conway's Game of Life on the 8x8 toroidal grid.

Original entry on oeis.org

4, 2, 4, 4, 4, 4, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 6, 4, 4, 2, 4, 4, 4, 4, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 6, 4, 4, 2, 4, 4, 4, 4, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 6, 4

Offset: 0

Antti Karttunen, Jul 27 2010

nonn

The mean value of terms in the whole period of 24 is 6.41667.

			See illustrations in A179409. In the first four patterns there are alive cells in 4, 2, 4 and 4 rows, thus a(0)=4, a(1)=2 and a(2)=a(3)=4.

A. Karttunen, J-program for computing the terms of this sequence

A179409-A179410.

A179412 The number of alive cells in Conway's Game of Life on the 8 X 8 toroidal grid, in a cyclic sequence of 132 patterns, whose initial pattern is given in illustrations below.

Original entry on oeis.org

8, 8, 9, 10, 12, 16, 13, 23, 16, 22, 18, 24, 16, 20, 21, 23, 28, 19, 18, 18, 23, 22, 18, 27, 16, 20, 10, 10, 10, 13, 15, 19, 22, 18, 25, 18, 19, 23, 23, 20, 21, 22, 30, 19, 22, 21, 20, 28, 19, 16, 14, 9, 13, 12, 13, 14, 16, 23, 15, 19, 16, 26, 16, 12, 12, 9, 8, 8, 9, 10, 12, 16

Offset: 0

Antti Karttunen, Jul 27 2010

nonn

Period 66. The sequence begins (from offset 0) with its lexicographically earliest rotation. Note the almost symmetric subsequence around the terms 66k and 66k+1: ...,16,12,12,9,8,8,9,10,12,16,... All integers in range [8,30] occur except 11, 17 and 29. The mean value of terms in the whole period of 66 is 17.7273.

This is the longest cyclic sequence that I have found so far (July 2010) on 8 X 8 toroidal grid, after the cycle of 48 given in A179409. Are there any longer cyclic sequences? A sequence to be computed: for n X n toroidal grid, the longest cycle of patterns that can occur. (Also other metrics for toroidal boards: how many patterns die in next generation, how many are stable, etc.)

			The generations 0-3 of this cycle of patterns look as follows, thus a(0)=a(1)=8, a(2)=9 and a(3)=10. Note how the initial pattern differs by just one misplaced cell from the pattern present in the generation 3 of A179409.
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(generation 0.) | (generation 1.) | (generation 2.) | (generation 3.)
In generation 66 we obtain a mirror image of the initial pattern, and in the generations 66--131 the patterns repeat the history of the first 66 generations, but reflected over the vertical axis, after which the whole cycle begins from the start again, at the generation 132.
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(generation 66) | (generation 67)

Antti Karttunen, J-program for computing the terms of this sequence
Antti Karttunen, Electronic music project using this Life-pattern sequence as one of its preferred sources of variation
Index entries for sequences related to music

Cf. A179413, A179414, A179409.

Cf. also A060118 (another input for the music project).

A179413 The number of columns with alive cells in the n-th generation of cyclic sequence of patterns given in A179412, played in Conway's Game of Life on the 8x8 toroidal grid.

Original entry on oeis.org

3, 4, 5, 5, 6, 6, 6, 7, 8, 8, 8, 8, 7, 8, 8, 8, 8, 8, 7, 7, 8, 8, 8, 7, 7, 6, 3, 5, 5, 5, 6, 8, 7, 8, 8, 7, 7, 7, 8, 7, 8, 8, 8, 7, 7, 8, 8, 8, 6, 6, 7, 5, 6, 8, 7, 7, 8, 8, 8, 6, 7, 7, 8, 6, 6, 4, 3, 4, 5, 5, 6, 6, 6, 7, 8, 8, 8, 8, 7, 8, 8, 8, 8, 8, 7, 7, 8, 8, 8, 7, 7, 6, 3, 5, 5, 5, 6, 8, 7

Offset: 0

Antti Karttunen, Jul 27 2010

nonn

Period 66. The sequence begins (from offset 0) with its lexicographically earliest rotation.

The mean value of terms in the whole period of 66 is 6.84848.

			See illustrations in A179412. In the first four patterns there are alive cells in 3, 4, 5, 5 and 5 columns, thus a(0)=3, a(1)=4, and a(2)=a(3)=5.

A. Karttunen, J-program for computing the terms of this sequence

A179414 The number of rows with alive cells in the n-th generation of cyclic sequence of patterns given in A179412, played in Conway's Game of Life on the 8x8 toroidal grid.

Original entry on oeis.org

4, 4, 4, 4, 5, 6, 7, 8, 8, 8, 7, 8, 7, 7, 7, 8, 8, 8, 7, 7, 7, 7, 6, 8, 8, 8, 7, 4, 6, 6, 6, 7, 7, 8, 8, 7, 8, 7, 8, 8, 8, 8, 8, 8, 8, 7, 8, 8, 6, 7, 7, 5, 4, 5, 5, 5, 6, 6, 7, 7, 8, 8, 7, 5, 5, 5, 4, 4, 4, 4, 5, 6, 7, 8, 8, 8, 7, 8, 7, 7, 7, 8, 8, 8, 7, 7, 7, 7, 6, 8, 8, 8, 7, 4, 6, 6, 6, 7, 7

Offset: 0

Antti Karttunen, Jul 27 2010

nonn

Period 66. The sequence begins (from offset 0) with its lexicographically earliest rotation.

The mean value of terms in the whole period of 66 is 6.72727.

			See illustrations in A179412. In all the first four patterns there are alive cells in four rows, thus a(0)=a(1)=a(2)=a(3)=4.

A. Karttunen, J-program for computing the terms of this sequence

A268857 The number of alive cells in Conway's Game of Life starting with the "R-pentomino" initial pattern.

Original entry on oeis.org

5, 6, 7, 9, 8, 9, 12, 11, 18, 11, 11, 10, 13, 16, 19, 19, 23, 25, 35, 25, 32, 27, 37, 30, 46, 39, 45, 30, 31, 29, 27, 32, 32, 39, 34, 29, 34, 31, 34, 36, 33, 31, 29, 34, 31, 42, 37, 36, 45, 48, 64, 45, 60, 50, 67, 58, 66, 68, 72, 72, 79, 75, 80, 67, 69, 73, 65, 56, 61, 52, 53, 60

Offset: 0

Jean-François Alcover, Feb 14 2016

nonn

Population stabilizes at 116 alive cells after generation 1102.

Jean-François Alcover, Table of n, a(n) for n = 0..1103
Eric Weisstein's MathWorld, Game of Life
Wikipedia, Conway's Game of Life

Cf. A179409.

(* example with a 100x100 grid *) m = 50; GameOfLife = {224, {2, {{2, 2, 2}, {2, 1, 2}, {2, 2, 2}}}, {1, 1}}; grids = CellularAutomaton[ GameOfLife, SparseArray[{{m-1, m} -> 1, {m-1, m+1} -> 1, {m, m-1} -> 1, {m, m} -> 1, {m+1, m} -> 1}, {2m, 2m}], 2m]; a[n_] := grids[[n+1]] // Flatten // Total; Table[a[n], {n, 0, 2 m}]

A179415 The number of alive cells in Conway's Game of Life on the infinite square grid, in the "lumps of muck" sequence of patterns leading from the grandfather of "stairstep hexomino" to a stable configuration of four blocks, known as a blockade.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jul 27 2010

nonn

For n >= 65, a(n)=16. Note that the same history is traced on any toroidal grid with size at least 26 X 26.

All terms are even because the initial pattern has an even number of cells and because it has 180-degree rotational symmetry.

			The generations 0-3 of this cycle of patterns look as follows, thus a(0)=a(1)=a(2)=6 and a(3)=8.
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..................................|stairstep hexomino................
Generations 4-7 of this cycle of patterns look as follows, thus a(4)=10, a(5)=12, a(6)=16 and a(7)=18.
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.(handshake). . | . . . . . . . . | . . . . . . . . | . . . . . . . .
At generation 65, the following stable formation of four blocks is reached, called "blockade", and thus for n >= 65, a(n)=16.
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Antti Karttunen, J-program for computing the terms of this sequence
Stephen A. Silver, Life Lexicon, "lumps of muck"

A179409, which traces the history of the same initial pattern on an 8 X 8 toroidal grid, differs from this one for the first time at n=14, as a(14)=32, while A179409(14)=26.

A268284 Period 15: repeat {18, 20, 28, 20, 20, 22, 18, 22, 20, 16, 12, 22, 18, 40, 18}.

Original entry on oeis.org

18, 20, 28, 20, 20, 22, 18, 22, 20, 16, 12, 22, 18, 40, 18, 18, 20, 28, 20, 20, 22, 18, 22, 20, 16, 12, 22, 18, 40, 18, 18, 20, 28, 20, 20, 22, 18, 22, 20, 16, 12, 22, 18, 40, 18, 18, 20, 28, 20, 20, 22, 18, 22, 20, 16, 12, 22, 18, 40, 18, 18, 20, 28, 20, 20, 22, 18, 22, 20, 16, 12, 22, 18, 40, 18

Offset: 0

Ilya Gutkovskiy, Jan 30 2016

Number of living cells periodic figure (oscillators: pentadecathlon (period 15)) in the Conway's Game of Life (rule B3/S23: see Graphical example in Links section).

			Start pattern (see Graphical example in Links section):
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|(generation 0)                 |(generation 1), etc.            |

Golly, Cross-platform application for exploring Conway's Game of Life and other cellular automata
Ilya Gutkovskiy, Graphical example (generation 0-15)
Eric Weisstein's World of Mathematics, Game of Life
Wikipedia, Conway's Game of Life
Wikipedia, Oscillator (cellular automaton)
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A061342, A152389, A179409.

&cat[[18,20,28,20,20,22,18,22,20,16,12,22,18,40,18]^^7]; // Vincenzo Librandi, Jan 30 2016

LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {18, 20, 28, 20, 20, 22, 18, 22, 20, 16, 12, 22, 18, 40, 18}, 80]

a(n)=2*[9, 10, 14, 10, 10, 11, 9, 11, 10, 8, 6, 11, 9, 20, 9][n%15+1] \\ Charles R Greathouse IV, Jul 17 2016

For k>=0:
a((30*k - 2*sin((Pi*k)/2) - 18*cos((Pi*k)/2) - cos(Pi*k) + 19)/8) = 18;
a((30*k + 10*sin((Pi*k)/2) + 18*cos((Pi*k)/2) + 3*cos(Pi*k) - 13)/8) = 20;
a(15*k + 2) = 28;
a(15*k + 9) = 16;
a(15*k + 10) = 12;
a(15*k + 13) = 40.

A268575 The number of alive cells in Conway's Game of Life starting with the "Die hard" initial pattern.

Original entry on oeis.org

7, 8, 8, 11, 10, 10, 12, 12, 16, 16, 24, 18, 17, 19, 19, 25, 21, 30, 19, 21, 18, 18, 22, 20, 20, 20, 23, 20, 23, 20, 15, 23, 15, 16, 16, 15, 16, 17, 20, 18, 19, 21, 25, 27, 30, 33, 30, 36, 30, 29, 24, 22, 22, 23, 23, 37, 26, 26, 28, 31, 33, 35, 30, 30, 25, 23, 17, 15, 12, 16, 12, 14, 11, 8, 9, 10, 13, 15, 22, 13, 15, 16, 17, 17, 24, 20, 25, 21, 29, 23, 27, 28, 37, 32, 40, 22, 30, 23, 21, 21, 23, 17, 21, 17, 16, 16, 20, 15, 23, 18, 21, 20, 23, 20, 22, 13, 15, 10, 13, 9, 11, 7, 5, 7, 6, 5, 6, 5, 3, 2, 0

Offset: 0

Jean-François Alcover, Feb 21 2016

Population vanishes after 130 generations.

LifeWiki, Die hard
Eric Weisstein's MathWorld, Game of Life
Wikipedia, Conway's Game of Life

Cf. A179409, A268857.

maxIter = 200; halfWidth = h = 100; GameOfLife = {224, {2, {{2, 2, 2}, {2, 1, 2}, {2, 2, 2}}}, {1, 1}}; grids = CellularAutomaton[GameOfLife, SparseArray[{{h + 3, h + 2} -> 1, {h - 3, h + 1} -> 1, {h - 2, h + 1} -> 1, {h - 2, h} -> 1, {h + 2, h} -> 1, {h + 3, h} -> 1, {h + 4, h} -> 1} ], 2h]; a[n_] := grids[[n + 1]] // Flatten // Total; Table[a[n], {n, 0, maxIter}] //. {a__, 0, 0} :> {a, 0}

cp's OEIS Frontend

A179410 The number of columns with alive cells in the n-th generation of cyclic sequence of patterns given in A179409, played in Conway's Game of Life on the 8x8 toroidal grid.

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A179411 The number of rows with alive cells in the n-th generation of cyclic sequence of patterns given in A179409, played in Conway's Game of Life on the 8x8 toroidal grid.

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A179412 The number of alive cells in Conway's Game of Life on the 8 X 8 toroidal grid, in a cyclic sequence of 132 patterns, whose initial pattern is given in illustrations below.

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A179413 The number of columns with alive cells in the n-th generation of cyclic sequence of patterns given in A179412, played in Conway's Game of Life on the 8x8 toroidal grid.

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A179414 The number of rows with alive cells in the n-th generation of cyclic sequence of patterns given in A179412, played in Conway's Game of Life on the 8x8 toroidal grid.

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A268857 The number of alive cells in Conway's Game of Life starting with the "R-pentomino" initial pattern.

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Mathematica

A179415 The number of alive cells in Conway's Game of Life on the infinite square grid, in the "lumps of muck" sequence of patterns leading from the grandfather of "stairstep hexomino" to a stable configuration of four blocks, known as a blockade.

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A268284 Period 15: repeat {18, 20, 28, 20, 20, 22, 18, 22, 20, 16, 12, 22, 18, 40, 18}.

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Magma

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PARI

Formula

A268575 The number of alive cells in Conway's Game of Life starting with the "Die hard" initial pattern.

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