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A279701
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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 185", based on the 5-celled von Neumann neighborhood.
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4
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1, 0, 5, 0, 29, 8, 85, 40, 261, 232, 1365, 296, 6725, 4264, 3349, 55976, 62789, 125608, 218389, 436904, 873541, 1747624, 3494165, 6990504, 13976901, 27962024, 55907605, 111848104, 223630405, 447392424, 894521621, 1789569704, 3578086725, 7158278824
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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 > 27.
G.f.: (65536*x^27 + 32768*x^26 + 98304*x^25 - 49152*x^24 - 38912*x^23 + 23552*x^22 - 3072*x^21 - 1280*x^20 - 64896*x^19 - 33088*x^18 - 98368*x^17 + 49472*x^16 + 38912*x^15 - 23520*x^14 + 3072*x^13 + 1256*x^12 - 632*x^11 + 320*x^10 + 72*x^9 - 80*x^8 + 8*x^7 - 31*x^6 + 8*x^5 + 9*x^4 + x^2 + 1)/(4*x^10 - x^8 - 4*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 = 185; 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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