A152389 -id:A152389 - cp's OEIS frontend

A098720 a(3) = 0; otherwise, a(n) = A152389(n) - 1.

0, 0, 0, 1, 5, 11, 13, 47, 19, 1, 14, 14, 23, 27, 39, 31, 23, 19, 24, 19, 18, 34, 29, 27, 92, 23, 27, 32, 35, 102, 147, 59, 579, 41, 56, 90, 105, 261, 275, 48, 208, 56, 51, 55, 96, 53, 167, 193, 810, 102, 51, 51, 82, 56, 78, 245, 415, 61, 61, 311, 114, 115

Offset: 1

Axel Harvey, Sep 29 2004

nonn

			After a single unique pattern consisting of a solid 3 by 8-cell block, the sequence of patterns following a 10-cell straight line becomes a cyclical process of repeating patterns. Therefore a(10) = 1.

LifeWiki, One cell thick pattern
Eric Weisstein's World of Mathematics, Game of Life

Cf. A152301, A152389.

a(21)-a(34) from John M. Campbell, May 05 2012

More terms and definition changed by Eric M. Schmidt, Aug 15 2012

A152301 a(3) = 1; otherwise a(n) = A152389(n).

Original entry on oeis.org

1, 1, 1, 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

Offset: 1

N. J. A. Sloane, Oct 23 2009

nonn

Cf. A098720, A152389.

More terms and definition changed by Eric M. Schmidt, Aug 15 2012

A061342 Period of the stationary component of the pattern which a row of n cells becomes in Conway's Game of Life.

Original entry on oeis.org

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

Offset: 1

Alex Fink, Jun 06 2001

nonn

For n = 56, the pattern produces gliders and thus does not oscillate in the truest sense of the word. For n = 57, no gliders are produced and a(57) = 2. There exists an N such that for all n >= N, the row of n alive cells produces gliders and thus does not oscillate. - Nathaniel Johnston, Jun 08 2009

			a(10) = 15 because a line of 10 cells becomes the pentadecathlon.
a(56) = 2 because a line of 56 cells becomes 4 gliders (which we ignore) and a remaining pattern of period 2. - _Eric M. Schmidt_, May 24 2014

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
LifeWiki, One-cell-thick pattern

Cf. A152389.

More general definition and sequence extended by Eric M. Schmidt, May 24 2014

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):
|. . . . . . . . . . . . . . . .| . . . . . . . . . . . . . . . .|
|. . . . . . . . . . . . . . . .| . . . . . . . . . . . . . . . .|
|. . . . o . . . . . . o . . . .| . . . o o . . . . . . o o . . .|
|. . . o o . . . . . . o o . . .| . . o . . o . . . . o . . o . .|
|. . o o o . . . . . . o o o . .| . . o . . o . . . . o . . o . .|
|. . . o o . . . . . . o o . . .| . . o . . o . . . . o . . o . .|
|. . . . o . . . . . . o . . . .| . . . o o . . . . . . o o . . .|
|. . . . . . . . . . . . . . . .| . . . . . . . . . . . . . . . .|
|. . . . . . . . . . . . . . . .| . . . . . . . . . . . . . . . .|
|(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.

A370776 The maximum number of alive cells reached in Conway's Game of Life when starting with the first n primes in Ulam's spiral; or -1 if no such maximum exists.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 65, 56, 120, 56, 28, 133, 30, 160, 46, 24, 24, 25, 28, 30, 31, 31, 32, 32, 32, 35, 74, 39, 38, 38, 38, 39, 40, 42, 319, 319, 319, 319, 319, 46, 129, 93, 50, 50, 72, 72, 72, 72, 72, 72, 53, 53, 56, 56, 851, 851, 167, 167, 167, 167, 391

Offset: 1

Thomas Strohmann, Mar 01 2024

nonn

The initial alive cells are at coordinates x=A214664(i), y=A214665(i) for i=1..n.
For the first 7 terms of this sequence we have a(n)=n since those initial configurations do not lead to complex enough patterns that increase the number of alive cells beyond the initial number of alive cells.
The definition includes the possibility that a glider gun (or a similar pattern) is created which will result in an unbounded number of alive cells.

			n=1 to n=4 die out very quickly (within 3 steps). The maximum number of alive cells is simply the number of alive cells in the initial pattern, i.e., n.
n=5 is the first term that leads to somewhat interesting steps in the game of life simulation (although the maximum number of alive cells still does not exceed the initial number 5):
  . . . . . | . . . . . | . . . o . | . . . o . | . . . o . | . . . . .
  o . o . . | . o o o . | . . o . o | . . o . o | . . . o . | . . . . .
  . . o o . | . . o o . | . . o . o | . . . . . | . . . . . | . . . . .
  o . . . . | . . . . . | . . . . . | . . . . . | . . . . . | . . . . .
n=8 leads to a maximum number of 65 alive cells and stabilizes after 107 steps. Initial pattern:
  o . . . o |
  . o . o . |
  o . . o o |
  . o . . . |
n=15 reaches a maximum of 160 alive cells and is the first pattern that leads to having a glider (escaping in the northeast direction). Besides the glider, the stabilized pattern contains 4 blinkers, 3 blocks, 2 beehives and 1 ship.

Wikipedia, Ulam spiral.
Wikipedia, Conway's Game of Life.

Cf. A214664, A214665.

Cf. A152389, A126803, A099733, A179412.

cp's OEIS Frontend

A098720 a(3) = 0; otherwise, a(n) = A152389(n) - 1.

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A152301 a(3) = 1; otherwise a(n) = A152389(n).

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A061342 Period of the stationary component of the pattern which a row of n cells becomes in Conway's Game of Life.

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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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A370776 The maximum number of alive cells reached in Conway's Game of Life when starting with the first n primes in Ulam's spiral; or -1 if no such maximum exists.

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