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proposed

Cf. A286859, A286860, A286862.

Robert Price, <a href="/A286861/a286861.tmp.txt">Diagrams of first 20 stages</a>

Robert Price, <a href="/A286861/a286861.tmp.txt">Diagrams of first 20 stages</a>

Robert Price, <a href="/A286861/b286861.txt">Table of n, a(n) for n = 0..126</a>

Robert Price, <a href="/A286861/a286861.tmp.txt">Diagrams of first 20 stages</a>

allocated for Robert PriceDecimal representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 813", based on the 5-celled von Neumann neighborhood.

1, 2, 7, 14, 31, 63, 127, 255, 511, 1008, 2047, 4095, 8191, 16383, 32767, 65535, 131071, 261919, 524287, 1048575, 2097151, 4194239, 8388607, 16777215, 33554431, 67108863, 134217727, 268435423, 536870911, 1073741775, 2147483647, 4294967039, 8589934591

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>

Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</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 = 813; 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}];

Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]

allocated

nonn,easy

Robert Price, Aug 01 2017

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allocated for Robert Price

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allocated for Shahryar Teymour Tash

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