Number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 137", based on the 5-celled von Neumann neighborhood.

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Robert Price, <a href="/A270274/a270274.tmp.txt">Diagrams of the first 20 stages.</a>

Robert Price, <a href="/A270274/a270274.tmp.txt">Diagrams of the first 20 stages.</a>

Robert Price, <a href="/A270274/b270274.txt">Table of n, a(n) for n = 0..128</a>

Robert Price, <a href="/A270274/a270274.tmp.txt">Diagrams of the first 20 stages.</a>

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allocated for Robert PriceNumber of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 137", based on the 5-celled von Neumann neighborhood.

1, 4, 5, 28, 9, 84, 29, 132, 37, 268, 37, 392, 69, 524, 101, 616, 141, 836, 145, 1104, 173, 1352, 229, 1584, 265, 1876, 329, 2180, 397, 2408, 521, 2576, 569, 2984, 633, 3484, 625, 4032, 657, 4500, 737, 5024, 789, 5612, 793, 6132, 929, 6508, 1029, 7072, 1141

0,2

Initialized with a single black (ON) cell at stage zero.

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

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

Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>

S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>

<a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>

<a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>

<a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>

CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];

code=137; stages=128;

rule=IntegerDigits[code, 2, 10];

g=2*stages+1; (* Maximum size of grid *)

a=PadLeft[{{1}}, {g, g}, 0, Floor[{g, g}/2]]; (* Initial ON cell on grid *)

ca=a;

ca=Table[ca=CAStep[rule, ca], {n, 1, stages+1}];

PrependTo[ca, a];

(* Trim full grid to reflect growth by one cell at each stage *)

k=(Length[ca[[1]]]+1)/2;

ca=Table[Table[Part[ca[[n]][[j]], Range[k+1-n, k-1+n]], {j, k+1-n, k-1+n}], {n, 1, k}];

Map[Function[Apply[Plus, Flatten[#1]]], ca] (* Count ON cells at each stage *)

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Robert Price, Mar 14 2016

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