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A284349
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Decimal representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 873", based on the 5-celled von Neumann neighborhood.
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4
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1, 0, 3, 5, 10, 29, 58, 95, 175, 503, 1019, 1525, 3066, 8189, 16382, 24575, 45055, 129023, 262143, 393215, 786431, 2097151, 4194303, 6291455, 11534335, 33030143, 67108863, 100663295, 201326591, 536870911, 1073741823, 1610612735, 2952790015, 8455716863
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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) = a(n-1) + 256*a(n-8) - 256*a(n-9) for n > 23.
G.f.: (-256*x^23 - 256*x^22 - 768*x^21 - 1280*x^20 + 1536*x^19 + 1024*x^18 - 1279*x^15 + 769*x^14 + 259*x^13 + 261*x^12 - 6*x^11 - 252*x^10 + 584*x^9 - 176*x^8 + 37*x^7 + 29*x^6 + 19*x^5 + 5*x^4 + 2*x^3 + 3*x^2 - x + 1)/(256*x^9 - 256*x^8 - x + 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 = 873; 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[1, i]], 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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