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A279501
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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 161", based on the 5-celled von Neumann neighborhood.
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4
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1, 0, 5, 0, 29, 0, 93, 0, 381, 0, 1373, 0, 5501, 0, 21853, 0, 87421, 0, 349533, 0, 1398141, 0, 5592413, 0, 22369661, 0, 89478493, 0, 357913981, 0, 1431655773, 0, 5726623101, 0, 22906492253, 0, 91625969021, 0, 366503875933, 0, 1466015503741, 0, 5864062014813
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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-4) - 4*a(n-6) for n>6.
G.f.: (1 + x^2 + 8*x^4 - 24*x^6 - 128*x^10) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)*(1 + x^2)).
(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 = 161; 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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