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A273831
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Number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 961", based on the 5-celled von Neumann neighborhood.
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4
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1, 4, 21, 45, 76, 117, 164, 221, 284, 357, 436, 525, 620, 725, 836, 957, 1084, 1221, 1364, 1517, 1676, 1845, 2020, 2205, 2396, 2597, 2804, 3021, 3244, 3477, 3716, 3965, 4220, 4485, 4756, 5037, 5324, 5621, 5924, 6237, 6556, 6885, 7220, 7565, 7916, 8277, 8644
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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) = (-7-(-1)^n+8*n+8*n^2)/2 for n>2.
a(n) = 4*(n^2+n-1) for n>2 and even.
a(n) = 4*n^2+4*n-3 for n>2 and odd.
a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) for n>6.
G.f.: (1+2*x+13*x^2+5*x^3-7*x^4+3*x^5-x^6) / ((1-x)^3*(1+x)).
(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=961; 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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