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A290415
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Decimal representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 785", based on the 5-celled von Neumann neighborhood.
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4
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1, 1, 5, 7, 19, 63, 7, 255, 15, 1023, 31, 4095, 63, 16383, 127, 65535, 255, 262143, 511, 1048575, 1023, 4194303, 2047, 16777215, 4095, 67108863, 8191, 268435455, 16383, 1073741823, 32767, 4294967295, 65535, 17179869183, 131071, 68719476735, 262143
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history;
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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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G.f.: (1 - 2*x^2 + 2*x^3 - 4*x^4 + 32*x^5 - 96*x^6 + 192*x^8 - 128*x^9) / ((1 - x)*(1 - 2*x)*(1 + 2*x)*(1 - 2*x^2)).
a(n) = a(n-1) + 6*a(n-2) - 6*a(n-3) - 8*a(n-4) + 8*a(n-5) for n>6.
(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 = 785; 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}]
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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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