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A282829
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Decimal representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 521", based on the 5-celled von Neumann neighborhood.
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4
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1, 0, 4, 0, 28, 56, 84, 0, 380, 248, 1364, 0, 5884, 3448, 22740, 13696, 91004, 50936, 363860, 202752, 1455868, 814456, 5822676, 3257728, 23290748, 13027064, 93162836, 52107264, 372651772, 208432504, 1490606292, 833729920, 5962425212, 3334915832, 23849700692
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OFFSET
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0,3
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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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a(n) = 4*a(n-2) + a(n-8) - 4*a(n-10) for n>15.
G.f.: (1 + 12*x^4 + 56*x^5 - 28*x^6 - 224*x^7 + 43*x^8 + 248*x^9 - 156*x^10 - 992*x^11 + 416*x^12 + 3392*x^13 - 768*x^14 + 128*x^15 - 4096*x^17) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)*(1 + x^2)*(1 + x^4)).
(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 = 521; 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]], 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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