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A273405
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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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3
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1, 4, 21, 44, 77, 116, 165, 220, 285, 356, 437, 524, 621, 724, 837, 956, 1085, 1220, 1365, 1516, 1677, 1844, 2021, 2204, 2397, 2596, 2805, 3020, 3245, 3476, 3717, 3964, 4221, 4484, 4757, 5036, 5325, 5620, 5925, 6236, 6557, 6884, 7221, 7564, 7917, 8276, 8645
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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)^n+8*n+8*n^2-7)/2 for n>0.
a(n) = 4*n^2+4*n-3 for n>0 and even.
a(n) = 4*(n^2+n-1) for n>0 and odd.
a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) for n>4.
G.f.: (1+2*x+13*x^2+4*x^3-4*x^4) / ((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=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}];
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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