cp's OEIS Frontend

A271257 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 324", based on the 5-celled von Neumann neighborhood.

%I A271257 #9 Feb 16 2025 08:33:32
%S A271257 1,5,10,22,30,54,79,119,152,196,236,324,401,501,622,798,935,1099,1240,
%T A271257 1444,1661,1949,2258,2642,2995,3399,3800,4308,4821,5385,5986,6710,
%U A271257 7411,8083,8812,9636,10485,11449,12430,13598,14727,15851,16964,18232,19573,20997
%N A271257 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 324", based on the 5-celled von Neumann neighborhood.
%C A271257 Initialized with a single black (ON) cell at stage zero.
%D A271257 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A271257 Robert Price, <a href="/A271257/b271257.txt">Table of n, a(n) for n = 0..128</a>
%H A271257 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 A271257 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A271257 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A271257 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A271257 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A271257 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%t A271257 CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
%t A271257 code=324; stages=128;
%t A271257 rule=IntegerDigits[code,2,10];
%t A271257 g=2*stages+1; (* Maximum size of grid *)
%t A271257 a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
%t A271257 ca=a;
%t A271257 ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
%t A271257 PrependTo[ca,a];
%t A271257 (* Trim full grid to reflect growth by one cell at each stage *)
%t A271257 k=(Length[ca[[1]]]+1)/2;
%t A271257 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 A271257 on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
%t A271257 Table[Total[Part[on,Range[1,i]]],{i,1,Length[on]}] (* Sum at each stage *)
%Y A271257 Cf. A246316.
%K A271257 nonn,easy
%O A271257 0,2
%A A271257 _Robert Price_, Apr 02 2016