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A269782
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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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5
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1, 4, 5, 36, 9, 96, 17, 188, 21, 312, 25, 468, 29, 656, 33, 876, 37, 1128, 41, 1412, 45, 1728, 49, 2076, 53, 2456, 57, 2868, 61, 3312, 65, 3788, 69, 4296, 73, 4836, 77, 5408, 81, 6012, 85, 6648, 89, 7316, 93, 8016, 97, 8748, 101, 9512, 105, 10308, 109, 11136
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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-(-1)^n+4*(-1)^n*n-4*(-1+(-1)^n)*n^2)/2 for n>4.
a(n) = 2*n+5 for n>4 and even.
a(n) = 4*n^2-2*n+6 for n>4 and odd.
a(n) = 3*a(n-2)-3*a(n-4)+a(n-6) 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)^3*(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}];
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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