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A271052
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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 253", based on the 5-celled von Neumann neighborhood.
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1
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1, 9, 13, 57, 70, 186, 199, 419, 432, 788, 801, 1325, 1338, 2062, 2075, 3031, 3044, 4264, 4277, 5793, 5806, 7650, 7663, 9867, 9880, 12476, 12489, 15509, 15522, 18998, 19011, 22975, 22988, 27472, 27485, 32521, 32534, 38154, 38167, 44403, 44416, 51300, 51313
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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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G.f.: (1 + 8*x + x^2 + 20*x^3 + 4*x^4 + 8*x^5 - 15*x^6 - 4*x^7 + 9*x^8) / ((1 - x)^4*(1 + x)^3).
a(n) = (8*n^3 + 12*n^2 + 46*n - 48) / 12 for n>1 and even.
a(n) = (8*n^3 + 36*n^2 + 94*n - 138) / 12 for n>1 and odd.
a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7) for n>8.
(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=253; 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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