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
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
KEYWORD
nonn,uned
AUTHOR
Paul Stoeber (pstoeber(AT)uni-potsdam.de), Oct 03 2005
STATUS
approved