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A269712
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Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 20", based on the 5-celled von Neumann neighborhood.
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0
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1, 4, 12, 28, 60, 124, 252, 508, 1020, 2044, 4092, 8188, 16380, 32764, 65532, 131068
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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.
Rules 28, 52, 60, 148, 156, 180, 188, 532, 540, 564, 572, 660, 668, 692 and 700 also generate this sequence.
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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) = 4*(2^n-1) =A028399(n+2) for n>0.
a(n) = 3*a(n-1)-2*a(n-2) for n>2.
G.f.: (1+x+2*x^2) / ((1-x)*(1-2*x)).
(End)
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MATHEMATICA
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rule=20; stages=300;
ca=CellularAutomaton[{rule, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}}, {1, 1}}, {{{1}}, 0}, stages]; (* Start with single black cell *)
on=Map[Function[Apply[Plus, Flatten[#1]]], ca] (* Count ON cells at each stage *)
Part[on, 2^Range[0, Log[2, stages]]] (* Extract relevant terms *)
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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