A055397 -id:A055397 - cp's OEIS frontend

A273308 Maximum population of a 2 X n still life in Conway's Game of Life.

0, 4, 4, 6, 8, 8, 10, 12, 12, 14, 16, 16, 18, 20, 20, 22, 24, 24, 26, 28, 28, 30, 32, 32, 34, 36, 36, 38, 40, 40, 42, 44, 44, 46, 48, 48, 50, 52, 52, 54, 56, 56, 58, 60, 60, 62, 64, 64, 66, 68, 68, 70, 72, 72, 74, 76, 76, 78, 80, 80, 82, 84, 84, 86, 88, 88

Offset: 1

Nathaniel Johnston, May 19 2016

Although the Chu et al. reference does not discuss this problem explicitly, the same methods in that paper can be used to prove the formula for this sequence.

			a(2) = 4 because the largest number of alive cells in a 2 X 2 still life is 4, which is attained by the block.
a(4) = 6 because the largest number of alive cells in a 2 X 4 still life is 6, which is attained by the snake.

Colin Barker, Table of n, a(n) for n = 1..1000
G. Chu, P. Stuckey, and M.G. de la Banda, Using relaxations in Maximum Density Still Life, In Proc. of Fifteenth Intl. Conf. on Principles and Practice of Constraint Programming, 258-273 (2009).
LifeWiki, Still life
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A019473, A055397.

seq(4*floor((n+1)*(1/3))+2*floor((1/2)*(`mod`(n+1, 3))), n = 2 .. 110);

LinearRecurrence[{1,0,1,-1},{0,4,4,6,8},70] (* Harvey P. Dale, Apr 19 2023 *)

concat(0, Vec(2*x^2*(2+x^2-x^3)/((1-x)^2*(1+x+x^2)) + O(x^50))) \\ Colin Barker, May 24 2016

def A273308(n): return n+sum(divmod(n,3)) if n > 1 else 0 # Chai Wah Wu, Jan 29 2023

For n >= 1, a(3*n) = a(3*n-1) = 4*n and a(3*n+1) = 4*n+2.
From Colin Barker, May 24 2016: (Start)
a(n) = a(n-1)+a(n-3)-a(n-4) for n>5.
G.f.: 2*x^2*(2+x^2-x^3) / ((1-x)^2*(1+x+x^2)). (End)
a(n) = A063224(n+1) = A063200(n+1) for n>1. - R. J. Mathar, May 27 2016

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.

Original entry on oeis.org

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

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

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.

			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.

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

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

{- 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

cp's OEIS Frontend

A273308 Maximum population of a 2 X n still life in Conway's Game of Life.

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