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A090634
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Start with the sequence [1, 1/2, 1/3, ..., 1/n]; form new sequence of n-1 terms by taking averages of successive terms; repeat until reach a single number F(n); a(n) = denominator of F(n).
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5
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1, 4, 12, 32, 80, 64, 448, 1024, 2304, 5120, 11264, 8192, 53248, 114688, 245760, 524288, 1114112, 262144, 4980736, 2097152, 3145728, 46137344, 96468992, 67108864, 419430400, 872415232, 1811939328, 3758096384, 7784628224, 5368709120, 33285996544, 68719476736
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OFFSET
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1,2
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COMMENTS
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a(n) is the denominator of the resistance of the n-dimensional cube between two adjacent nodes, when the resistance of each edge is 1. See Nedermeyer and Smorodinsky. - Michel Marcus, Sep 13 2019
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LINKS
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FORMULA
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a(n) = denominator(2*(1-1/2^n)/n) (conjectured). - Michel Marcus, Sep 12 2019
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EXAMPLE
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n=3: [1, 1/2, 1/3] -> [3/4, 5/6] -> [7/12], so F(3) = 7/12. Sequence of F(n)'s begins 1, 3/4, 7/12, 15/32, 31/80, 21/64, 127/448, 255/1024, ...
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MAPLE
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a:= n-> denom(coeff(series(2*log((x/2-1)/(x-1)), x, n+1), x, n)):
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MATHEMATICA
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f[s_list] := Table[(s[[k]] + s[[k+1]])/2, {k, 1, Length[s]-1}];
a[n_] := Nest[f, 1/Range[n], n-1] // First // Denominator;
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PROG
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(Haskell)
import Data.Ratio (denominator, (%))
a090634 n = denominator z where
[z] = (until ((== 1) . length) avg) $ map (1 %) [1..n]
avg xs = zipWith (\x x' -> (x + x') / 2) (tail xs) xs
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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STATUS
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approved
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