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A273336 Partial sums of the 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 A273336 #12 Feb 16 2025 08:33:35
%S A273336 1,5,22,70,150,270,438,662,950,1310,1750,2278,2902,3630,4470,5430,
%T A273336 6518,7742,9110,10630,12310,14158,16182,18390,20790,23390,26198,29222,
%U A273336 32470,35950,39670,43638,47862,52350,57110,62150,67478,73102,79030,85270,91830,98718
%N A273336 Partial sums of the 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 A273336 Initialized with a single black (ON) cell at stage zero.
%D A273336 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A273336 Robert Price, <a href="/A273336/b273336.txt">Table of n, a(n) for n = 0..128</a>
%H A273336 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 A273336 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A273336 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A273336 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A273336 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A273336 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A273336 Conjectures from _Colin Barker_, May 20 2016: (Start)
%F A273336 a(n) = 2/3*(2*n^3+6*n^2+4*n-15) for n>1.
%F A273336 a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>5.
%F A273336 G.f.: (1+x+8*x^2+8*x^3-17*x^4+7*x^5) / (1-x)^4.
%F A273336 (End)
%t A273336 CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
%t A273336 code=657; stages=128;
%t A273336 rule=IntegerDigits[code,2,10];
%t A273336 g=2*stages+1; (* Maximum size of grid *)
%t A273336 a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
%t A273336 ca=a;
%t A273336 ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
%t A273336 PrependTo[ca,a];
%t A273336 (* Trim full grid to reflect growth by one cell at each stage *)
%t A273336 k=(Length[ca[[1]]]+1)/2;
%t A273336 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 A273336 on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
%t A273336 Table[Total[Part[on,Range[1,i]]],{i,1,Length[on]}] (* Sum at each stage *)
%Y A273336 Cf. A273334.
%K A273336 nonn,easy
%O A273336 0,2
%A A273336 _Robert Price_, May 20 2016