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A266285
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Number of ON (black) cells in the n-th iteration of the "Rule 13" elementary cellular automaton starting with a single ON (black) cell.
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1
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1, 1, 2, 4, 3, 7, 4, 10, 5, 13, 6, 16, 7, 19, 8, 22, 9, 25, 10, 28, 11, 31, 12, 34, 13, 37, 14, 40, 15, 43, 16, 46, 17, 49, 18, 52, 19, 55, 20, 58, 21, 61, 22, 64, 23, 67, 24, 70, 25, 73, 26, 76, 27, 79, 28, 82, 29, 85, 30, 88, 31, 91, 32, 94, 33, 97, 34
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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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Conjectures from Colin Barker, Dec 28 2015 and Apr 14 2019: (Start)
a(n) = ((-1)^n*(3-2*n)+4*n+1)/4.
a(n) = 2*a(n-2)-a(n-4) for n>3.
G.f.: (1+x+2*x^3) / ((1-x)^2*(1+x)^2).
(End)
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MATHEMATICA
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rule=13; 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 *) Table[Total[catri[[k]]], {k, 1, rows}] (* Number of Black cells in stage n *)
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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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EXTENSIONS
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STATUS
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approved
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