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A271060
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Number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 261", based on the 5-celled von Neumann neighborhood.
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4
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1, 8, 1, 48, 1, 120, 1, 224, 1, 360, 1, 528, 1, 728, 1, 960, 1, 1224, 1, 1520, 1, 1848, 1, 2208, 1, 2600, 1, 3024, 1, 3480, 1, 3968, 1, 4488, 1, 5040, 1, 5624, 1, 6240, 1, 6888, 1, 7568, 1, 8280, 1, 9024, 1, 9800, 1, 10608, 1, 11448, 1, 12320, 1, 13224, 1
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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) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n > 5.
G.f.: (-x^4 - 24*x^3 + 2*x^2 - 8*x - 1)/((x - 1)^3*(x + 1)^3). (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=261; 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}];
Map[Function[Apply[Plus, Flatten[#1]]], ca] (* Count ON cells at each stage *)
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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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