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A272742 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 A272742 #11 Feb 16 2025 08:33:34
%S A272742 1,5,9,17,17,33,45,69,61,85,97,121,121,169,205,277,245,293,305,329,
%T A272742 329,377,413,485,461,533,569,641,641,785,893,1109,1005,1125,1137,1161,
%U A272742 1161,1209,1245,1317,1293,1365,1401,1473,1473,1617,1725,1941,1845,1989
%N A272742 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 A272742 Initialized with a single black (ON) cell at stage zero.
%D A272742 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A272742 Robert Price, <a href="/A272742/b272742.txt">Table of n, a(n) for n = 0..128</a>
%H A272742 Robert Price, <a href="/A272742/a272742.tmp.txt">Diagrams of the first 20 stages</a>
%H A272742 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 A272742 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A272742 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A272742 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A272742 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A272742 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%t A272742 CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
%t A272742 code=526; stages=128;
%t A272742 rule=IntegerDigits[code,2,10];
%t A272742 g=2*stages+1; (* Maximum size of grid *)
%t A272742 a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
%t A272742 ca=a;
%t A272742 ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
%t A272742 PrependTo[ca,a];
%t A272742 (* Trim full grid to reflect growth by one cell at each stage *)
%t A272742 k=(Length[ca[[1]]]+1)/2;
%t A272742 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 A272742 Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
%K A272742 nonn,easy
%O A272742 0,2
%A A272742 _Robert Price_, May 05 2016