This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A384694 #28 Jul 23 2025 16:09:12 %S A384694 0,0,3,12,35,92,228,544,1264,2880,6464,14336,31488,68608,148480, %T A384694 319488,684032,1458176,3096576,6553600,13828096,29097984,61079552, %U A384694 127926272,267386880,557842432,1161822208,2415919104,5016387584,10401873920,21541945344,44560285696,92073361408,190052302848,391915765760 %N 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. %H A384694 LifeWiki, <a href="https://www.conwaylife.com/wiki/One-cell-thick_pattern">One cell thick pattern</a> %H A384694 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4). %F A384694 G.f.: x^2*(3 - x^2)/(1 - 4*x + 4*x^2). %F A384694 a(n) = 2^(n - 5) * (11*n - 20). %F A384694 E.g.f.: (9 - 4*x - 2*x^2 + exp(2*x)*(22*x - 9))/16. - _Stefano Spezia_, Jun 07 2025 %e A384694 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. %Y A384694 Cf. A167667 (after one generation). %K A384694 nonn,easy %O A384694 0,3 %A A384694 _SiYang Hu_, Jun 07 2025