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A273861 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 1006", based on the 5-celled von Neumann neighborhood.

%I A273861 #10 Feb 16 2025 08:33:36
%S A273861 1,6,15,36,65,106,167,256,357,470,603,764,953,1186,1495,1872,2269,
%T A273861 2678,3107,3564,4049,4578,5183,5856,6565,7318,8139,9044,10049,11218,
%U A273861 12631,14200,15805,17422,19059,20724,22417,24154,25967,27848,29765,31726,33755,35868
%N A273861 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 1006", based on the 5-celled von Neumann neighborhood.
%C A273861 Initialized with a single black (ON) cell at stage zero.
%D A273861 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A273861 Robert Price, <a href="/A273861/b273861.txt">Table of n, a(n) for n = 0..128</a>
%H A273861 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 A273861 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A273861 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A273861 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A273861 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A273861 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%t A273861 CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
%t A273861 code=1006; stages=128;
%t A273861 rule=IntegerDigits[code,2,10];
%t A273861 g=2*stages+1; (* Maximum size of grid *)
%t A273861 a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
%t A273861 ca=a;
%t A273861 ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
%t A273861 PrependTo[ca,a];
%t A273861 (* Trim full grid to reflect growth by one cell at each stage *)
%t A273861 k=(Length[ca[[1]]]+1)/2;
%t A273861 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 A273861 on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
%t A273861 Table[Total[Part[on,Range[1,i]]],{i,1,Length[on]}] (* Sum at each stage *)
%Y A273861 Cf. A169709.
%K A273861 nonn,easy
%O A273861 0,2
%A A273861 _Robert Price_, Jun 01 2016