cp's OEIS Frontend

A272700 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 510", based on the 5-celled von Neumann neighborhood.

%I A272700 #9 Feb 16 2025 08:33:34
%S A272700 1,6,18,43,71,127,183,296,356,476,596,836,956,1196,1436,1917,2041,
%T A272700 2289,2537,3033,3281,3777,4273,5265,5513,6009,6505,7497,7993,8985,
%U A272700 9977,11962,12214,12718,13222,14230,14734,15742,16750,18766,19270,20278,21286,23302
%N A272700 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 510", based on the 5-celled von Neumann neighborhood.
%C A272700 Initialized with a single black (ON) cell at stage zero.
%D A272700 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A272700 Robert Price, <a href="/A272700/b272700.txt">Table of n, a(n) for n = 0..128</a>
%H A272700 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 A272700 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A272700 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A272700 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A272700 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A272700 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%t A272700 CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
%t A272700 code=510; stages=128;
%t A272700 rule=IntegerDigits[code,2,10];
%t A272700 g=2*stages+1; (* Maximum size of grid *)
%t A272700 a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
%t A272700 ca=a;
%t A272700 ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
%t A272700 PrependTo[ca,a];
%t A272700 (* Trim full grid to reflect growth by one cell at each stage *)
%t A272700 k=(Length[ca[[1]]]+1)/2;
%t A272700 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 A272700 on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
%t A272700 Table[Total[Part[on,Range[1,i]]],{i,1,Length[on]}] (* Sum at each stage *)
%Y A272700 Cf. A169699.
%K A272700 nonn,easy
%O A272700 0,2
%A A272700 _Robert Price_, May 04 2016