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
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).
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)
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
KEYWORD
nonn,easy
AUTHOR
Nathaniel Johnston, May 19 2016
STATUS
approved