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A273419
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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 705", based on the 5-celled von Neumann neighborhood.
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1
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1, 5, 26, 67, 139, 252, 412, 629, 909, 1262, 1694, 2215, 2831, 3552, 4384, 5337, 6417, 7634, 8994, 10507, 12179, 14020, 16036, 18237, 20629, 23222, 26022, 29039, 32279, 35752, 39464, 43425, 47641, 52122, 56874, 61907, 67227, 72844, 78764, 84997, 91549, 98430
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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) = (111-3*(-1)^n-58*n+48*n^2+16*n^3)/12 for n>1.
a(n) = (8*n^3+24*n^2-29*n+54)/6 for n>1 and even.
a(n) = (8*n^3+24*n^2-29*n+57)/6 for n>1 and odd.
a(n) = 3*a(n-1)-2*a(n-2)-2*a(n-3)+3*a(n-4)-a(n-5) for n>6.
G.f.: (1+2*x+13*x^2+x^3-3*x^4+7*x^5-5*x^6) / ((1-x)^4*(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=705; 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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