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A270085
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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 65", based on the 5-celled von Neumann neighborhood.
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1
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1, 5, 10, 46, 55, 151, 168, 356, 377, 689, 714, 1182, 1211, 1867, 1900, 2776, 2813, 3941, 3982, 5394, 5439, 7167, 7216, 9292, 9345, 11801, 11858, 14726, 14787, 18099, 18164, 21952, 22021, 26317, 26390, 31226, 31303, 36711, 36792, 42804, 42889, 49537, 49626
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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) = 1/4*(-45+(-1)^n)+(35*n)/6-(-1+(-1)^n)*n^2+(2*n^3)/3 for n>3.
a(n) = (4*n^3+35*n-66)/6 for n>3 and even.
a(n) = (4*n^3+12*n^2+35*n-69)/6 for n>3 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.
G.f.: (1+4*x+2*x^2+24*x^3-3*x^4+4*x^6+4*x^7-8*x^8+4*x^10) / ((1-x)^4*(1+x)^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=65; 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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