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A272422 First differences of number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 470", based on the 5-celled von Neumann neighborhood.

%I A272422 #11 Feb 16 2025 08:33:34
%S A272422 4,3,16,-3,35,-24,57,-24,75,-44,105,-52,111,-60,97,-4,83,-8,85,8,103,
%T A272422 -32,153,-20,139,-40,165,-16,139,8,85,32,119,109,0,-1,268,-151,324,
%U A272422 -149,356,-143,308,-65,264,-55,168,115,132,37,240,-145,408,-223,368,51
%N A272422 First differences of number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 470", based on the 5-celled von Neumann neighborhood.
%C A272422 Initialized with a single black (ON) cell at stage zero.
%D A272422 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A272422 Robert Price, <a href="/A272422/b272422.txt">Table of n, a(n) for n = 0..127</a>
%H A272422 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 A272422 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A272422 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A272422 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A272422 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A272422 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%t A272422 CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
%t A272422 code=470; stages=128;
%t A272422 rule=IntegerDigits[code,2,10];
%t A272422 g=2*stages+1; (* Maximum size of grid *)
%t A272422 a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
%t A272422 ca=a;
%t A272422 ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
%t A272422 PrependTo[ca,a];
%t A272422 (* Trim full grid to reflect growth by one cell at each stage *)
%t A272422 k=(Length[ca[[1]]]+1)/2;
%t A272422 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 A272422 on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
%t A272422 Table[on[[i+1]]-on[[i]],{i,1,Length[on]-1}] (* Difference at each stage *)
%Y A272422 Cf. A253078.
%K A272422 sign,easy
%O A272422 0,1
%A A272422 _Robert Price_, Apr 29 2016