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%I A270106 #10 Jul 26 2024 21:16:32
%S A270106 1,5,13,29,45,77,109,173,205,269,333,461,525,653,781,1037,1101,1229,
%T A270106 1357,1613,1741,1997,2253,2765,2893,3149,3405,3917,4173,4685,5197,
%U A270106 6221,6349,6605,6861,7373,7629,8141,8653,9677,9933,10445,10957,11981,12493,13517
%N A270106 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 84", based on the 5-celled von Neumann neighborhood.
%C A270106 Initialized with a single black (ON) cell at stage zero.
%D A270106 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A270106 Robert Price, <a href="/A270106/b270106.txt">Table of n, a(n) for n = 0..128</a>
%H A270106 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 A270106 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A270106 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A270106 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A270106 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A270106 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%t A270106 CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
%t A270106 code=84; stages=128;
%t A270106 rule=IntegerDigits[code,2,10];
%t A270106 g=2*stages+1; (* Maximum size of grid *)
%t A270106 a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
%t A270106 ca=a;
%t A270106 ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
%t A270106 PrependTo[ca,a];
%t A270106 (* Trim full grid to reflect growth by one cell at each stage *)
%t A270106 k=(Length[ca[[1]]]+1)/2;
%t A270106 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 A270106 on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
%t A270106 Table[Total[Part[on,Range[1,i]]],{i,1,Length[on]}] (* Sum at each stage *)
%Y A270106 Cf. A189007.
%K A270106 nonn,easy
%O A270106 0,2
%A A270106 _Robert Price_, Mar 11 2016

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