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A267354
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Number of OFF (white) cells in the n-th iteration of the "Rule 123" elementary cellular automaton starting with a single ON (black) cell.
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1
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0, 1, 2, 1, 6, 1, 10, 1, 14, 1, 18, 1, 22, 1, 26, 1, 30, 1, 34, 1, 38, 1, 42, 1, 46, 1, 50, 1, 54, 1, 58, 1, 62, 1, 66, 1, 70, 1, 74, 1, 78, 1, 82, 1, 86, 1, 90, 1, 94, 1, 98, 1, 102, 1, 106, 1, 110, 1, 114, 1, 118, 1, 122, 1, 126, 1, 130, 1, 134, 1, 138, 1
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OFFSET
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0,3
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REFERENCES
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S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
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LINKS
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FORMULA
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a(n) = ((2*(-1)^n+2)*n-3*(-1)^n-1)/2 for n>0.
a(n) = 2*a(n-2)-a(n-4) for n>4.
G.f.: x*(1+2*x-x^2+2*x^3) / ((1-x)^2*(1+x)^2).
(End)
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MATHEMATICA
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rule=123; rows=20; ca=CellularAutomaton[rule, {{1}, 0}, rows-1, {All, All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]], {rows-k+1, rows+k-1}], {k, 1, rows}]; (* Truncated list of each row *) nbc=Table[Total[catri[[k]]], {k, 1, rows}]; (* Number of Black cells in stage n *) Table[Length[catri[[k]]]-nbc[[k]], {k, 1, rows}] (* Number of White cells in stage n *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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