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A272421 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 470", based on the 5-celled von Neumann neighborhood.

%I A272421 #11 Feb 16 2025 08:33:34
%S A272421 1,6,14,38,59,115,147,236,301,441,537,738,887,1147,1347,1644,1937,
%T A272421 2313,2681,3134,3595,4159,4691,5376,6041,6845,7609,8538,9451,10503,
%U A272421 11563,12708,13885,15181,16586,17991,19395,21067,22588,24433,26129,28181,30090,32307
%N A272421 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 470", based on the 5-celled von Neumann neighborhood.
%C A272421 Initialized with a single black (ON) cell at stage zero.
%D A272421 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A272421 Robert Price, <a href="/A272421/b272421.txt">Table of n, a(n) for n = 0..128</a>
%H A272421 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 A272421 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A272421 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A272421 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A272421 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A272421 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%t A272421 CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
%t A272421 code=470; stages=128;
%t A272421 rule=IntegerDigits[code,2,10];
%t A272421 g=2*stages+1; (* Maximum size of grid *)
%t A272421 a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
%t A272421 ca=a;
%t A272421 ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
%t A272421 PrependTo[ca,a];
%t A272421 (* Trim full grid to reflect growth by one cell at each stage *)
%t A272421 k=(Length[ca[[1]]]+1)/2;
%t A272421 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 A272421 on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
%t A272421 Table[Total[Part[on,Range[1,i]]],{i,1,Length[on]}] (* Sum at each stage *)
%Y A272421 Cf. A253078.
%K A272421 nonn,easy
%O A272421 0,2
%A A272421 _Robert Price_, Apr 29 2016