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A273386
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Partial sums of the number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 659", based on the 5-celled von Neumann neighborhood.
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1
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1, 6, 23, 72, 153, 274, 443, 668, 957, 1318, 1759, 2288, 2913, 3642, 4483, 5444, 6533, 7758, 9127, 10648, 12329, 14178, 16203, 18412, 20813, 23414, 26223, 29248, 32497, 35978, 39699, 43668, 47893, 52382, 57143, 62184, 67513, 73138, 79067, 85308, 91869, 98758
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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) = (11*n)/3+4*n^2+(4*n^3)/3-11 for n>1.
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>5.
G.f.: (1+2*x+5*x^2+12*x^3-20*x^4+8*x^5) / (1-x)^4.
(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=659; 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}];
on=Map[Function[Apply[Plus, Flatten[#1]]], ca] (* Count ON cells at each stage *)
Table[Total[Part[on, Range[1, i]]], {i, 1, Length[on]}] (* Sum 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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