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A278661
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Decimal representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 57", based on the 5-celled von Neumann neighborhood.
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4
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1, 0, 1, 12, 5, 48, 7, 244, 25, 976, 7, 4084, 113, 16244, 21, 65472, 415, 261388, 97, 1048428, 1985, 4192028, 577, 16775196, 5057, 67093020, 6357, 268419616, 23183, 1073684512, 4015, 4294962736, 68279, 17179631792, 69175, 68719306412, 321389, 274876908426
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OFFSET
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0,4
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COMMENTS
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Initialized with a single black (ON) cell at stage zero.
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REFERENCES
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S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
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LINKS
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Robert Price, Table of n, a(n) for n = 0..126
Robert Price, Diagrams of first 20 stages
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata
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MATHEMATICA
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CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=57; 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[1, i]], 2], {i, 1, stages-1}]
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CROSSREFS
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Cf. A278659, A278660, A278662.
Sequence in context: A110185 A299516 A278423 * A280614 A281041 A038331
Adjacent sequences: A278658 A278659 A278660 * A278662 A278663 A278664
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KEYWORD
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nonn,easy
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AUTHOR
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Robert Price, Nov 25 2016
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STATUS
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approved
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