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A273407
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First differences of number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 673", based on the 5-celled von Neumann neighborhood.
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2
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3, 17, 23, 33, 39, 49, 55, 65, 71, 81, 87, 97, 103, 113, 119, 129, 135, 145, 151, 161, 167, 177, 183, 193, 199, 209, 215, 225, 231, 241, 247, 257, 263, 273, 279, 289, 295, 305, 311, 321, 327, 337, 343, 353, 359, 369, 375, 385, 391, 401, 407, 417, 423, 433
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OFFSET
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0,1
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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) = 8-(-1)^n+8*n for n>0.
a(n) = 8*n+7 for n>0 and even.
a(n) = 8*n+9 for n>0 and odd.
a(n) = a(n-1)+a(n-2)-a(n-3) for n>3.
G.f.: (3+14*x+3*x^2-4*x^3) / ((1-x)^2*(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=673; 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[on[[i+1]]-on[[i]], {i, 1, Length[on]-1}] (* Difference 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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