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A270129
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Number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 89", based on the 5-celled von Neumann neighborhood.
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4
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1, 4, 5, 44, 0, 121, 0, 225, 0, 361, 0, 529, 0, 729, 0, 961, 0, 1225, 0, 1521, 0, 1849, 0, 2209, 0, 2601, 0, 3025, 0, 3481, 0, 3969, 0, 4489, 0, 5041, 0, 5625, 0, 6241, 0, 6889, 0, 7569, 0, 8281, 0, 9025, 0, 9801, 0, 10609, 0, 11449, 0, 12321, 0, 13225, 0
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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) = -1/2*(-1+(-1)^n)*(1+2*n)^2 for n>3.
a(n) = 0 for n>3 and n even.
a(n) = 4*n^2+4*n+1 for n>3 and n odd.
a(n) = 3*a(n-2)-3*a(n-4)+a(n-6) for n>9.
G.f.: (1+4*x+2*x^2+32*x^3-12*x^4+x^5+14*x^6-10*x^7-5*x^8+5*x^9) / ((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=89; 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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