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A283583
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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 646", based on the 5-celled von Neumann neighborhood.
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4
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1, 3, 5, 13, 21, 53, 81, 219, 321, 867, 1349, 3437, 5445, 13677, 20805, 56173, 82245, 222061, 345413, 879981, 1393989, 3501421, 5326149, 14380397, 21054789, 56847725, 88425797, 225275245, 356861253, 896363885, 1363494213, 3681381741, 5390026053, 14553017709
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OFFSET
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0,2
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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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FORMULA
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G.f.: (1 + 3*x + 4*x^2 + 10*x^3 + 16*x^4 + 40*x^5 + 60*x^6 + 166*x^7 - 16*x^8 - 120*x^9 + 4*x^10 + 10*x^11) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)*(1 + 4*x^2)*(1 + 16*x^4)).
a(n) = a(n-2) + 256*a(n-8) - 256*a(n-10) for n>11.
(End)
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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 = 646; 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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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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