A061342 -id:A061342 - cp's OEIS frontend

A152389 Number of steps in Conway's Game of Life for a row of n cells to stabilize.

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

Offset: 0

N. J. A. Sloane, Oct 23 2009, based on a posting by Allan C. Wechsler to the Math Fun Mailing List

nonn

A pattern is said to have stabilized if it consists entirely of a (possibly empty) periodic component and zero or more spaceships, such that the spaceships will never interact with each other or with the periodic part.

			From _Eric M. Schmidt_, Aug 15 2012: (Start)
A 10-cell straight line evolves into a periodic pattern (the pentadecathlon) in two steps. Therefore a(10) = 2. (Based on example in A098720)
A 33-cell straight line evolves, in 387 steps, into a pattern consisting of a periodic component and four gliders. The pattern has not yet stabilized since the gliders will eventually collide.
A 56-cell straight line evolves, in 246 steps, into a pattern consisting of a periodic component and four gliders. The gliders will never collide with each other or with the periodic component, so the pattern has stabilized. Thus, a(56) = 246. (End)

Eric M. Schmidt, Table of n, a(n) for n = 0..1000
LifeWiki, One cell thick pattern
Eric M. Schmidt, C++ code to compute this sequence
Eric Weisstein's World of Mathematics, Game of Life

Cf. A098720, A152301.

Cf. A061342.

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

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.

A110910 Configurations in the evolution of a line of n cells in Conway's Game of Life, with 0=infinity. For periodic evolutions, a(n)=(preperiod length)+(period length). For non-periodic evolutions, a(n)=0.

Original entry on oeis.org

1, 2, 2, 2, 3, 8, 13, 15, 49, 22, 17, 17, 16, 26, 29, 41, 34, 25, 21, 26, 21, 21, 36, 31, 29, 95, 25, 29, 34, 38, 105, 150, 61, 582, 43, 58, 92, 108, 263, 277, 50, 212, 59, 53, 57, 99, 55, 170, 196, 812, 105, 54, 53, 85, 59, 81, 0, 418, 63, 63, 314, 117, 118, 170, 236, 104

Offset: 0

Paul Stoeber (pstoeber(AT)uni-potsdam.de), Oct 03 2005

If nothing catches up with an outbound glider, then a(n)=0 for n>=1000 because when you watch the horizontal 1000-line evolve in a simulator, around the 490th generation, gliders fly away from the left and right corners before the non-chaotic growing in the middle has finished, so you will see the same local picture in the 490th generation of longer lines.

			a(0)=1 because there is only the empty configuration. a(10)=2+15 because the 10-line needs two steps to become a pentadecathlon. a(56)=0 because the 56-line sends four gliders to outer space.

Berlekamp/Conway/Guy, Winning Ways ..., 2nd ed, vol. 4, chapter 25

Cf. A061342, A019473, A056605, A056614, A055397, A099733, A089520, A098720, A056613.

{- program for verification of periodic cases. The non-periodic cases listed here evolve into a periodic kernel plus gliders whose paths ahead do not intersect each other or the kernel (gliders marching in single file are not counted as intersecting). -}
import Data.Set
main = print [if n `elem` known then 0 else a n | n<-[0..105]]
known = [56, 71, 72, 75, 78, 82, 85, 86, 87, 88, 91, 92, 93, 94, 96, 98, 100, 102, 103, 105]
a n = count empty (iterate evolve (fromList [(x, 0) | x<-[1..n]]))
neighbors (x, y) = fromList
                  [(x+u, y+v) | u<-[ -1, 0, 1], v<-[ -1, 0, 1], (u, v)/=(0, 0)]
evolve life =
  let fil f = Data.Set.filter
              (\x-> f (size (life `intersection` neighbors x)))
  in (life `difference` fil (\k-> k<2 || k>3) life) `union` fil (== 3)
     (unions (Prelude.map neighbors (elems life)) `difference` life)
count o (x:xs) | x `member` o = 0
               | otherwise = 1 + count (o `union` singleton x) xs

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A152389 Number of steps in Conway's Game of Life for a row of n cells to stabilize.

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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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A110910 Configurations in the evolution of a line of n cells in Conway's Game of Life, with 0=infinity. For periodic evolutions, a(n)=(preperiod length)+(period length). For non-periodic evolutions, a(n)=0.

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