OFFSET
0,3
LINKS
Paolo Xausa, Table of n, a(n) for n = 0..1000
LifeWiki, One cell thick pattern
Index entries for linear recurrences with constant coefficients, signature (4,-4).
FORMULA
G.f.: x^2*(3 - x^2)/(1 - 4*x + 4*x^2).
a(n) = 2^(n - 4) * (11*n - 9) for n >= 3.
E.g.f.: (9 - 4*x - 2*x^2 + exp(2*x)*(22*x - 9))/16. - Stefano Spezia, Jun 07 2025
EXAMPLE
For n = 5, there are 5 ways for the cells to evolve into a blinker: ..OOO, O.OOO, .OOO., OOO.., OOO.O; 4 ways for the cells to evolve into a beehive predecessor and then a beehive: OOOO., .OOOO; 1 way for it to evolve into 8 cells: OOOOO, so a(5) = 3 * 5 + 6 * 2 + 8 * 1 = 35.
MATHEMATICA
LinearRecurrence[{4, -4}, {0, 0, 3, 12, 35}, 35] (* Paolo Xausa, Jan 11 2026 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
SiYang Hu, Jun 07 2025
STATUS
approved
