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A278446
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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 43", based on the 5-celled von Neumann neighborhood.
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4
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1, 2, 4, 11, 20, 43, 80, 175, 336, 687, 1344, 2751, 5440, 10943, 21760, 43775, 87296, 174847, 349184, 699391, 1397760, 2796543, 5591040, 11186175, 22368256, 44740607, 89473024, 178962431, 357908480, 715833343, 1431633920, 2863333375, 5726601216, 11453267967
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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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Empirical g.f.: (1 + 2*x) * (1 - 2*x + 3*x^2 - 3*x^3) / ((1 - x) * (1 + x) * (1 - 2*x) * (1 - 2*x^2) * (1 + 2*x^2)). - Colin Barker, Nov 23 2016
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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=43; 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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