

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 nonperiodic evolutions, a(n)=0.


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
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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 1000line evolve in a simulator, around the 490th generation, gliders fly away from the left and right corners _before_ the nonchaotic 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 10line needs two steps to become a pentadecathlon. a(56)=0 because the 56line sends four gliders to outer space.


PROG

{ Haskell program for verification of periodic cases. The
nonperiodic 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 by 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: A101360 A270371 A184311 * A119532 A010583 A238188
Adjacent sequences: A110907 A110908 A110909 * A110911 A110912 A110913


KEYWORD

nonn,uned


AUTHOR

Paul Stoeber (pstoeber(AT)unipotsdam.de), Oct 03 2005


STATUS

approved



