A384694 Sum of the number of cells alive after 2 generations of Conway's game of life for initial 1 X n cells taken in all 2^n combinations of alive or dead.

0, 0, 3, 12, 35, 92, 228, 544, 1264, 2880, 6464, 14336, 31488, 68608, 148480, 319488, 684032, 1458176, 3096576, 6553600, 13828096, 29097984, 61079552, 127926272, 267386880, 557842432, 1161822208, 2415919104, 5016387584, 10401873920, 21541945344, 44560285696, 92073361408, 190052302848, 391915765760

Offset: 0

Author

SiYang Hu, Jun 07 2025

Keywords

• nonn
• easy

Examples

			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.
		

Links

• LifeWiki, One cell thick pattern
• Index entries for linear recurrences with constant coefficients, signature (4,-4).

Crossrefs

Cf. A167667 (after one generation).

Formula

G.f.: x^2*(3 - x^2)/(1 - 4*x + 4*x^2).

a(n) = 2^(n - 5) * (11*n - 20).

E.g.f.: (9 - 4*x - 2*x^2 + exp(2*x)*(22*x - 9))/16. - Stefano Spezia, Jun 07 2025