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 A265283 #29 Feb 16 2025 08:33:27
%S A265283 1,3,4,6,6,8,8,10,10,12,12,14,14,16,16,18,18,20,20,22,22,24,24,26,26,
%T A265283 28,28,30,30,32,32,34,34,36,36,38,38,40,40,42,42,44,44,46,46,48,48,50,
%U A265283 50,52,52,54,54,56,56,58,58,60,60,62,62,64,64,66,66,68
%N A265283 Number of ON (black) cells in the n-th iteration of the "Rule 94" elementary cellular automaton starting with a single ON (black) cell.
%C A265283 From _Gus Wiseman_, Apr 13 2019: (Start)
%C A265283 Also the number of integer partitions of n + 3 such that lesser of the maximum part and the number of parts is 2. The Heinz numbers of these partitions are given by A325229. For example, the a(0) = 1 through a(7) = 10 partitions are:
%C A265283 (21) (22) (32) (33) (43) (44) (54) (55)
%C A265283 (31) (41) (42) (52) (53) (63) (64)
%C A265283 (211) (221) (51) (61) (62) (72) (73)
%C A265283 (2111) (222) (2221) (71) (81) (82)
%C A265283 (2211) (22111) (2222) (22221) (91)
%C A265283 (21111) (211111) (22211) (222111) (22222)
%C A265283 (221111) (2211111) (222211)
%C A265283 (2111111) (21111111) (2221111)
%C A265283 (22111111)
%C A265283 (211111111)
%C A265283 (End)
%D A265283 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
%H A265283 Robert Price, <a href="/A265283/b265283.txt">Table of n, a(n) for n = 0..999</a>
%H A265283 FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000533">St000533: The maximal number of non-attacking rooks on a Ferrers shape</a>
%H A265283 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A265283 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A265283 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A265283 Conjectures from _Colin Barker_, Dec 07 2015 and Apr 16 2019: (Start)
%F A265283 a(n) = (5-(-1)^n+2*n)/2 = A213222(n+3) for n>1.
%F A265283 a(n) = n+2 for n>1 and even.
%F A265283 a(n) = n+3 for n>1 and odd.
%F A265283 a(n) = a(n-1) + a(n-2) - a(n-3) for n>2.
%F A265283 G.f.: (1+2*x-x^4) / ((1-x)^2*(1+x)).
%F A265283 (End)
%e A265283 From _Michael De Vlieger_, Dec 14 2015: (Start)
%e A265283 First 12 rows, replacing "0" with "." for better visibility of ON cells, followed by the total number of 1's per row:
%e A265283 1 = 1
%e A265283 1 1 1 = 3
%e A265283 1 1 . 1 1 = 4
%e A265283 1 1 1 . 1 1 1 = 6
%e A265283 1 1 . 1 . 1 . 1 1 = 6
%e A265283 1 1 1 . 1 . 1 . 1 1 1 = 8
%e A265283 1 1 . 1 . 1 . 1 . 1 . 1 1 = 8
%e A265283 1 1 1 . 1 . 1 . 1 . 1 . 1 1 1 = 10
%e A265283 1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1 = 10
%e A265283 1 1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1 1 = 12
%e A265283 1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1 = 12
%e A265283 1 1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1 1 = 14
%e A265283 1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1 = 14
%e A265283 (End)
%t A265283 rule = 94; rows = 30; Table[Total[Table[Take[CellularAutomaton[rule, {{1},0},rows-1,{All,All}][[k]], {rows-k+1, rows+k-1}], {k,1,rows}][[k]]], {k,1,rows}]
%t A265283 Total /@ CellularAutomaton[94, {{1}, 0}, 65] (* _Michael De Vlieger_, Dec 14 2015 *)
%Y A265283 Column k = 2 of A325227.
%Y A265283 Cf. A004526, A051924, A118102, A263297, A325193, A325225, A325228, A325229, A325232.
%K A265283 nonn,easy
%O A265283 0,2
%A A265283 _Robert Price_, Dec 06 2015