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A265225 Total number of ON (black) cells after n iterations of the "Rule 54" elementary cellular automaton starting with a single ON (black) cell.

%I A265225 #48 Feb 16 2025 08:33:27
%S A265225 1,4,6,12,15,24,28,40,45,60,66,84,91,112,120,144,153,180,190,220,231,
%T A265225 264,276,312,325,364,378,420,435,480,496,544,561,612,630,684,703,760,
%U A265225 780,840,861,924,946,1012,1035,1104,1128,1200,1225,1300,1326,1404,1431
%N A265225 Total number of ON (black) cells after n iterations of the "Rule 54" elementary cellular automaton starting with a single ON (black) cell.
%C A265225 Take the first 2n positive integers and choose n of them such that their sum: a) is divisible by n, and b) is minimal. It seems their sum equals a(n). - _Ivan N. Ianakiev_, Feb 16 2019
%D A265225 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
%H A265225 Robert Price, <a href="/A265225/b265225.txt">Table of n, a(n) for n = 0..999</a>
%H A265225 Emanuele Munarini, <a href="https://doi.org/10.4418/2021.76.1.14">Topological indices for the antiregular graphs</a>, Le Mathematiche (2021) Vol. 76, No. 1, see p. 301.
%H A265225 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A265225 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A265225 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A265225 Conjectures from _Colin Barker_, Dec 08 2015 and Apr 20 2019: (Start)
%F A265225 a(n) = (n+1)*(2*n -(-1)^n +5)/4.
%F A265225 a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.
%F A265225 G.f.: (1+3*x) / ((1-x)^3*(1+x)^2).
%F A265225 (End)
%F A265225 a(n) = n + 1 + (n+1) * floor((n+1)/2), conjectured. - _Wesley Ivan Hurt_, Dec 25 2016
%F A265225 a(n) = A093353(n) + n + 1, conjectured. - _Matej Veselovac_, Jan 21 2020
%e A265225 From _Michael De Vlieger_, Dec 14 2015: (Start)
%e A265225 First 12 rows, replacing "0" with "." for better visibility of ON cells, followed by the total number of 1's per row, and the running total up to that row:
%e A265225                         1                          =  1 ->  1
%e A265225                       1 1 1                        =  3 ->  4
%e A265225                     1 . . . 1                      =  2 ->  6
%e A265225                   1 1 1 . 1 1 1                    =  6 -> 12
%e A265225                 1 . . . 1 . . . 1                  =  3 -> 15
%e A265225               1 1 1 . 1 1 1 . 1 1 1                =  9 -> 24
%e A265225             1 . . . 1 . . . 1 . . . 1              =  4 -> 28
%e A265225           1 1 1 . 1 1 1 . 1 1 1 . 1 1 1            = 12 -> 40
%e A265225         1 . . . 1 . . . 1 . . . 1 . . . 1          =  5 -> 45
%e A265225       1 1 1 . 1 1 1 . 1 1 1 . 1 1 1 . 1 1 1        = 15 -> 60
%e A265225     1 . . . 1 . . . 1 . . . 1 . . . 1 . . . 1      =  6 -> 66
%e A265225   1 1 1 . 1 1 1 . 1 1 1 . 1 1 1 . 1 1 1 . 1 1 1    = 18 -> 84
%e A265225 1 . . . 1 . . . 1 . . . 1 . . . 1 . . . 1 . . . 1  =  7 -> 91
%e A265225 (End)
%p A265225 A265225:=n->1/4*(n+1)*(2*n-(-1)^n+5): seq(A265225(n), n=0..60); # _Wesley Ivan Hurt_, Dec 25 2016
%t A265225 rule = 54; rows = 30; Table[Total[Take[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}],k]],{k,1,rows}]
%t A265225 Accumulate[Total /@ CellularAutomaton[54, {{1}, 0}, 52]]
%Y A265225 Cf. A071030, A118108, A118109, A133872.
%K A265225 nonn,easy
%O A265225 0,2
%A A265225 _Robert Price_, Dec 05 2015