cp's OEIS Frontend

A273329 First differences of number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 646", based on the 5-celled von Neumann neighborhood.

%I A273329 #10 Feb 16 2025 08:33:35
%S A273329 4,0,12,-8,20,-12,44,-44,60,-44,92,-80,116,-92,188,-196,220,-196,268,
%T A273329 -256,292,-268,364,-364,412,-364,508,-472,580,-508,796,-828,876,-828,
%U A273329 972,-960,996,-972,1068,-1068,1116,-1068,1212,-1176,1284,-1212,1500,-1524
%N A273329 First differences of number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 646", based on the 5-celled von Neumann neighborhood.
%C A273329 Initialized with a single black (ON) cell at stage zero.
%D A273329 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A273329 Robert Price, <a href="/A273329/b273329.txt">Table of n, a(n) for n = 0..127</a>
%H A273329 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 A273329 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A273329 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A273329 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A273329 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A273329 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%t A273329 CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
%t A273329 code=646; stages=128;
%t A273329 rule=IntegerDigits[code,2,10];
%t A273329 g=2*stages+1; (* Maximum size of grid *)
%t A273329 a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
%t A273329 ca=a;
%t A273329 ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
%t A273329 PrependTo[ca,a];
%t A273329 (* Trim full grid to reflect growth by one cell at each stage *)
%t A273329 k=(Length[ca[[1]]]+1)/2;
%t A273329 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 A273329 on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
%t A273329 Table[on[[i+1]]-on[[i]],{i,1,Length[on]-1}] (* Difference at each stage *)
%Y A273329 Cf. A273326.
%K A273329 sign,easy
%O A273329 0,1
%A A273329 _Robert Price_, May 20 2016