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A273408 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 675", based on the 5-celled von Neumann neighborhood.

%I A273408 #19 Feb 16 2025 08:33:35
%S A273408 1,6,27,72,149,266,431,652,937,1294,1731,2256,2877,3602,4439,5396,
%T A273408 6481,7702,9067,10584,12261,14106,16127,18332,20729,23326,26131,29152,
%U A273408 32397,35874,39591,43556,47777,52262,57019,62056,67381,73002,78927,85164,91721,98606
%N A273408 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 675", based on the 5-celled von Neumann neighborhood.
%C A273408 Initialized with a single black (ON) cell at stage zero.
%D A273408 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A273408 Robert Price, <a href="/A273408/b273408.txt">Table of n, a(n) for n = 0..128</a>
%H A273408 Robert Price, <a href="/A273408/a273408.tmp.txt">Diagrams of the first 20 stages</a>
%H A273408 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 A273408 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A273408 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A273408 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A273408 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A273408 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A273408 Conjectures from _Colin Barker_, May 22 2016: (Start)
%F A273408 a(n) = (4*n^3+12*n^2-n+3)/3.
%F A273408 a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>3.
%F A273408 G.f.: (1+2*x+9*x^2-4*x^3) / (1-x)^4. (End)
%t A273408 CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
%t A273408 code=675; stages=128;
%t A273408 rule=IntegerDigits[code,2,10];
%t A273408 g=2*stages+1; (* Maximum size of grid *)
%t A273408 a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
%t A273408 ca=a;
%t A273408 ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
%t A273408 PrependTo[ca,a];
%t A273408 (* Trim full grid to reflect growth by one cell at each stage *)
%t A273408 k=(Length[ca[[1]]]+1)/2;
%t A273408 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 A273408 on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
%t A273408 Table[Total[Part[on,Range[1,i]]],{i,1,Length[on]}] (* Sum at each stage *)
%Y A273408 Cf. A078371.
%K A273408 nonn,easy
%O A273408 0,2
%A A273408 _Robert Price_, May 21 2016