A318858 Number of still-lifes synthesizable from n gliders in Conway's game of Life.
0, 6, 11, 111, 114, 217
Offset: 1
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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.
{- 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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