A090634 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).

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

Offset: 1

Author

N. J. A. Sloane, Dec 13 2003

Keywords

• nonn
• frac

Comments

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

Examples

			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, ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..3312 (first 200 terms from T. D. Noe)
• F. Nedermeyer and Y. Smorodinsky, Resistance in the multidimensional cube, Quantum, Sept/October 1996, pp. 12-15 (beware file is 75Mb).
• Putnam Competition, Problem B2, Solutions, 2003.

Crossrefs

Cf. A090633 (numerators).

Programs

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
-- Reinhard Zumkeller, Dec 08 2011

Maple

a:= n-> denom(coeff(series(2*log((x/2-1)/(x-1)), x, n+1), x, n)):
seq(a(n), n=1..35);  # Alois P. Heinz, Aug 02 2018

Mathematica

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;
Array[a, 40] (* Jean-François Alcover, Aug 02 2018 *)

Formula

a(n) = A131135(n)/2. - Paul Barry, Jun 17 2007

a(n) = denominator(2*(1-1/2^n)/n) (conjectured). - Michel Marcus, Sep 12 2019