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A282987
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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 541", based on the 5-celled von Neumann neighborhood.
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4
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1, 1, 1, 9, 9, 9, 105, 137, 105, 777, 1513, 1929, 1129, 9993, 9705, 10121, 103529, 153353, 67049, 911241, 1119337, 644873, 6620649, 8710025, 6558825, 53729033, 94045673, 80275337, 107746409, 858248969, 1505166825, 1284827017, 1725174889, 13728470793
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OFFSET
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0,4
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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) = 17*a(n-4) - 16*a(n-8) for n > 33.
G.f.: (-2097152*x^33 - 1572864*x^32 + 59506688*x^31 + 12320768*x^30 - 44826624*x^29 + 14155776*x^28 - 53215232*x^27 - 17432576*x^26 + 45219840*x^25 - 10813440*x^24 - 6619136*x^23 + 5636096*x^22 - 1802240*x^21 - 622592*x^20 + 770048*x^19 - 73728*x^18 - 4096*x^17 + 86016*x^16 - 20480*x^15 - 14336*x^14 - 3072*x^13 - 512*x^12 - 256*x^11 - 256*x^10 + 640*x^9 - 32*x^8 - 16*x^7 + 88*x^6 - 8*x^5 - 8*x^4 + 9*x^3 + x^2 + x + 1)/(16*x^8 - 17*x^4 + 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 = 541; 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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