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A270217 Number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 129", based on the 5-celled von Neumann neighborhood.

%I A270217 #11 Feb 16 2025 08:33:31
%S A270217 1,4,5,28,9,84,21,140,25,260,37,380,49,532,85,620,89,868,101,1116,113,
%T A270217 1396,149,1612,161,1956,197,2236,233,2548,341,2604,345,3108,357,3612,
%U A270217 369,4148,405,4620,417,5220,453,5756,489,6324,597,6636,609,7364,645
%N A270217 Number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 129", based on the 5-celled von Neumann neighborhood.
%C A270217 Initialized with a single black (ON) cell at stage zero.
%D A270217 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A270217 Robert Price, <a href="/A270217/b270217.txt">Table of n, a(n) for n = 0..128</a>
%H A270217 Robert Price, <a href="/A270217/a270217.tmp.txt">Diagrams of the first 20 stages.</a>
%H A270217 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 A270217 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A270217 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A270217 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A270217 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A270217 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%t A270217 CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
%t A270217 code=129; stages=128;
%t A270217 rule=IntegerDigits[code,2,10];
%t A270217 g=2*stages+1; (* Maximum size of grid *)
%t A270217 a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
%t A270217 ca=a;
%t A270217 ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
%t A270217 PrependTo[ca,a];
%t A270217 (* Trim full grid to reflect growth by one cell at each stage *)
%t A270217 k=(Length[ca[[1]]]+1)/2;
%t A270217 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 A270217 Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
%K A270217 nonn,easy
%O A270217 0,2
%A A270217 _Robert Price_, Mar 13 2016