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A281738
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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 390", based on the 5-celled von Neumann neighborhood.
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4
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1, 3, 1, 7, 17, 39, 65, 199, 65, 455, 1025, 2055, 4097, 8199, 16385, 32775, 65537, 131079, 262145, 524295, 1048577, 2097159, 4194305, 12582919, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7
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history;
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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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a(n) = a(n-2) for n > 25.
G.f.: (12582912*x^25 + 4194304*x^24 - 10485760*x^23 - 3145728*x^22 - 1572864*x^21 - 786432*x^20 - 393216*x^19 - 196608*x^18 - 98304*x^17 - 49152*x^16 - 24576*x^15 - 12288*x^14 - 6144*x^13 - 3072*x^12 - 1600*x^11 - 960*x^10 - 256*x^9 - 160*x^7 - 48*x^6 - 32*x^5 - 16*x^4 - 4*x^3 - 3*x - 1)/(x^2 - 1). (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 = 390; 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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