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A273308
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Maximum population of a 2 X n still life in Conway's Game of Life.
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2
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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FORMULA
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For n >= 1, a(3*n) = a(3*n-1) = 4*n and a(3*n+1) = 4*n+2.
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)
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EXAMPLE
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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.
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MAPLE
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seq(4*floor((n+1)*(1/3))+2*floor((1/2)*(`mod`(n+1, 3))), n = 2 .. 110);
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MATHEMATICA
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LinearRecurrence[{1, 0, 1, -1}, {0, 4, 4, 6, 8}, 70] (* Harvey P. Dale, Apr 19 2023 *)
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PROG
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(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)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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