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Search: id:A110910
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| A110910 |
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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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+0
1
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| 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
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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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.
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REFERENCES
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Berlekamp/Conway/Guy, Winning Ways ..., 2nd ed, vol. 4, chapter 25
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EXAMPLE
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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.
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PROGRAM
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{- 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 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
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CROSSREFS
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Cf. A061342 A019473 A056605 A056614 A055397 A099733 A089520 A098720 A056613.
Sequence in context: A091228 A134890 A101360 this_sequence A119532 A010583 A051007
Adjacent sequences: A110907 A110908 A110909 this_sequence A110911 A110912 A110913
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KEYWORD
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nonn,uned
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AUTHOR
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Paul Stoeber (pstoeber(AT)uni-potsdam.de), Oct 03 2005
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