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A273334 Number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 657", based on the 5-celled von Neumann neighborhood.

%I A273334 #13 Feb 16 2025 08:33:35
%S A273334 1,4,17,48,80,120,168,224,288,360,440,528,624,728,840,960,1088,1224,
%T A273334 1368,1520,1680,1848,2024,2208,2400,2600,2808,3024,3248,3480,3720,
%U A273334 3968,4224,4488,4760,5040,5328,5624,5928,6240,6560,6888,7224,7568,7920,8280,8648
%N A273334 Number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 657", based on the 5-celled von Neumann neighborhood.
%C A273334 Initialized with a single black (ON) cell at stage zero.
%D A273334 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A273334 Robert Price, <a href="/A273334/b273334.txt">Table of n, a(n) for n = 0..128</a>
%H A273334 Robert Price, <a href="/A273334/a273334.tmp.txt">Diagrams of the first 20 stages</a>
%H A273334 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 A273334 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A273334 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A273334 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A273334 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A273334 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A273334 Conjectures from _Colin Barker_, May 20 2016: (Start)
%F A273334 a(n) = 4*n*(1+n) for n>2.
%F A273334 a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>5.
%F A273334 G.f.: (1+x+8*x^2+8*x^3-17*x^4+7*x^5) / (1-x)^3.
%F A273334 (End)
%t A273334 CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
%t A273334 code=657; stages=128;
%t A273334 rule=IntegerDigits[code,2,10];
%t A273334 g=2*stages+1; (* Maximum size of grid *)
%t A273334 a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
%t A273334 ca=a;
%t A273334 ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
%t A273334 PrependTo[ca,a];
%t A273334 (* Trim full grid to reflect growth by one cell at each stage *)
%t A273334 k=(Length[ca[[1]]]+1)/2;
%t A273334 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 A273334 Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
%K A273334 nonn,easy
%O A273334 0,2
%A A273334 _Robert Price_, May 20 2016