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A273314 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 643", based on the 5-celled von Neumann neighborhood.

%I A273314 #12 Feb 16 2025 08:33:35
%S A273314 1,6,23,64,137,250,411,628,909,1262,1695,2216,2833,3554,4387,5340,
%T A273314 6421,7638,8999,10512,12185,14026,16043,18244,20637,23230,26031,29048,
%U A273314 32289,35762,39475,43436,47653,52134,56887,61920,67241,72858,78779,85012,91565,98446
%N A273314 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 643", based on the 5-celled von Neumann neighborhood.
%C A273314 Initialized with a single black (ON) cell at stage zero.
%D A273314 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A273314 Robert Price, <a href="/A273314/b273314.txt">Table of n, a(n) for n = 0..128</a>
%H A273314 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 A273314 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A273314 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A273314 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A273314 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A273314 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A273314 Conjectures from _Colin Barker_, May 19 2016: (Start)
%F A273314 a(n) = (4*n^3+12*n^2-13*n+15)/3 for n>0.
%F A273314 a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>4.
%F A273314 G.f.: (1+2*x+5*x^2+4*x^3-4*x^4) / (1-x)^4.
%F A273314 (End)
%t A273314 CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
%t A273314 code=643; stages=128;
%t A273314 rule=IntegerDigits[code,2,10];
%t A273314 g=2*stages+1; (* Maximum size of grid *)
%t A273314 a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
%t A273314 ca=a;
%t A273314 ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
%t A273314 PrependTo[ca,a];
%t A273314 (* Trim full grid to reflect growth by one cell at each stage *)
%t A273314 k=(Length[ca[[1]]]+1)/2;
%t A273314 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 A273314 on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
%t A273314 Table[Total[Part[on,Range[1,i]]],{i,1,Length[on]}] (* Sum at each stage *)
%Y A273314 Cf. A166147.
%K A273314 nonn,easy
%O A273314 0,2
%A A273314 _Robert Price_, May 19 2016