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A273790 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 931", based on the 5-celled von Neumann neighborhood.

%I A273790 #14 Feb 16 2025 08:33:36
%S A273790 1,6,31,80,161,282,451,676,965,1326,1767,2296,2921,3650,4491,5452,
%T A273790 6541,7766,9135,10656,12337,14186,16211,18420,20821,23422,26231,29256,
%U A273790 32505,35986,39707,43676,47901,52390,57151,62192,67521,73146,79075,85316,91877,98766
%N A273790 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 931", based on the 5-celled von Neumann neighborhood.
%C A273790 Initialized with a single black (ON) cell at stage zero.
%D A273790 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A273790 Robert Price, <a href="/A273790/b273790.txt">Table of n, a(n) for n = 0..128</a>
%H A273790 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 A273790 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>.
%H A273790 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>.
%H A273790 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A273790 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>.
%H A273790 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>.
%F A273790 Conjectures from _Colin Barker_, May 31 2016: (Start)
%F A273790 a(n) = (4*n^3+12*n^2+11*n-9)/3 for n>0.
%F A273790 a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>4.
%F A273790 G.f.: (1+2*x+13*x^2-12*x^3+4*x^4) / (1-x)^4. (End)
%F A273790 Conjectured e.g.f.: 4 + exp(x)*(4*x^3/3 + 8*x^2 + 9*x - 3). - _Stefano Spezia_, Dec 30 2024
%t A273790 CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
%t A273790 code=931; stages=128;
%t A273790 rule=IntegerDigits[code,2,10];
%t A273790 g=2*stages+1; (* Maximum size of grid *)
%t A273790 a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
%t A273790 ca=a;
%t A273790 ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
%t A273790 PrependTo[ca,a];
%t A273790 (* Trim full grid to reflect growth by one cell at each stage *)
%t A273790 k=(Length[ca[[1]]]+1)/2;
%t A273790 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 A273790 on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
%t A273790 Table[Total[Part[on,Range[1,i]]],{i,1,Length[on]}] (* Sum at each stage *)
%Y A273790 Cf. A273789.
%K A273790 nonn,easy
%O A273790 0,2
%A A273790 _Robert Price_, May 30 2016