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

1,2

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

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.

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

Example

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, ...

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 *)

Prog

(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

Crossrefs

Cf. A090633 (numerators).

Sequence in context: A118885 A097392 A373046 * A260186 A085750 A001787

Adjacent sequences: A090631 A090632 A090633 * A090635 A090636 A090637

Keyword

nonn,frac

Author

N. J. A. Sloane, Dec 13 2003

Status

approved