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A272744 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 526", based on the 5-celled von Neumann neighborhood.

%I A272744 #10 Feb 16 2025 08:33:34
%S A272744 1,6,15,32,49,82,127,196,257,342,439,560,681,850,1055,1332,1577,1870,
%T A272744 2175,2504,2833,3210,3623,4108,4569,5102,5671,6312,6953,7738,8631,
%U A272744 9740,10745,11870,13007,14168,15329,16538,17783,19100,20393,21758,23159,24632
%N A272744 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 526", based on the 5-celled von Neumann neighborhood.
%C A272744 Initialized with a single black (ON) cell at stage zero.
%D A272744 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A272744 Robert Price, <a href="/A272744/b272744.txt">Table of n, a(n) for n = 0..128</a>
%H A272744 N. J. A. Sloane, <a href="http://arxiv.org/abs/1503.01168">On the Number of ON Cells in Cellular Automata</a>, arXiv:1503.01168 [math.CO], 2015
%H A272744 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A272744 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A272744 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A272744 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A272744 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%t A272744 CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
%t A272744 code=526; stages=128;
%t A272744 rule=IntegerDigits[code,2,10];
%t A272744 g=2*stages+1; (* Maximum size of grid *)
%t A272744 a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
%t A272744 ca=a;
%t A272744 ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
%t A272744 PrependTo[ca,a];
%t A272744 (* Trim full grid to reflect growth by one cell at each stage *)
%t A272744 k=(Length[ca[[1]]]+1)/2;
%t A272744 ca=Table[Table[Part[ca[[n]][[j]],Range[k+1-n,k-1+n]],{j,k+1-n,k-1+n}],{n,1,k}];
%t A272744 on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
%t A272744 Table[Total[Part[on,Range[1,i]]],{i,1,Length[on]}] (* Sum at each stage *)
%Y A272744 Cf. A272742.
%K A272744 nonn,easy
%O A272744 0,2
%A A272744 _Robert Price_, May 05 2016