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A273308
Maximum population of a 2 X n still life in Conway's Game of Life.
2
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
1,2
COMMENTS
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.
LINKS
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
FORMULA
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
EXAMPLE
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.
MAPLE
seq(4*floor((n+1)*(1/3))+2*floor((1/2)*(`mod`(n+1, 3))), n = 2 .. 110);
MATHEMATICA
LinearRecurrence[{1, 0, 1, -1}, {0, 4, 4, 6, 8}, 70] (* Harvey P. Dale, Apr 19 2023 *)
PROG
(PARI) 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
(Python)
def A273308(n): return n+sum(divmod(n, 3)) if n > 1 else 0 # Chai Wah Wu, Jan 29 2023
CROSSREFS
Sequence in context: A292755 A294021 A200267 * A105790 A338895 A090682
KEYWORD
nonn,easy
AUTHOR
Nathaniel Johnston, May 19 2016
STATUS
approved