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A269908
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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 1", based on the 5-celled von Neumann neighborhood.
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1
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1, 5, 6, 50, 51, 167, 168, 388, 389, 745, 746, 1270, 1271, 1995, 1996, 2952, 2953, 4173, 4174, 5690, 5691, 7535, 7536, 9740, 9741, 12337, 12338, 15358, 15359, 18835, 18836, 22800, 22801, 27285, 27286, 32322, 32323, 37943, 37944, 44180, 44181, 51065, 51066
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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+9*(-1)^n-2*(1+12*(-1)^n)*n-12*(-2+(-1)^n)*n^2+8*n^3)/12.
a(n) = (4*n^3+6*n^2-13*n+6)/6 for n even.
a(n) = (4*n^3+18*n^2+11*n-3)/6 for n 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>6.
G.f.: (1+4*x-2*x^2+32*x^3+x^4-4*x^5) / ((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=1; 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,changed
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AUTHOR
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STATUS
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approved
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