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.

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

0,2

Comments

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.

References

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

Links

Table of n, a(n) for n=0..65.

Example

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.

Prog

(Haskell)
{- 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). Replace leading dots with spaces before running! -}
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

Crossrefs

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

Sequence in context: A351108 A378298 A184311 * A119532 A010583 A306343

Adjacent sequences: A110907 A110908 A110909 * A110911 A110912 A110913

Keyword

nonn,uned

Author

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

Status

approved