A090633 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) = numerator of F(n).

1, 3, 7, 15, 31, 21, 127, 255, 511, 1023, 2047, 1365, 8191, 16383, 32767, 65535, 131071, 29127, 524287, 209715, 299593, 4194303, 8388607, 5592405, 33554431, 67108863, 134217727, 268435455, 536870911, 357913941, 2147483647, 4294967295, 8589934591, 17179869183

Offset: 1

Author

N. J. A. Sloane, Dec 13 2003

Keywords

• nonn
• frac

Comments

F(n) is the resistance across a single resistor of an n-dimensional hypercube made of 1 ohm resistors. - Peter J. C. Moses, May 27 2004

Also, numerators of BINOMIAL transform of sequence [1, 1/2, 1/3, 1/4, ...]. - Gary W. Adamson, Apr 26 2005

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, ...
		

References

• Putnam Competition, 2003, Problem B2.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..3320 (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).

Crossrefs

Cf. A090634 (denominators).

Programs

Haskell

import Data.Ratio (numerator, (%))
a090633 n = numerator 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

f := proc(L) local t1,i; t1 := []; for i from 1 to nops(L)-1 do t1 := [op(t1), (L[i]+L[i+1])/2]; od: t1; end; f2 := n->[seq(1/i,i=1..n)];
F := proc(n) local L,i; L := f2(n); for i from 1 to n-1 do L := f(L); od: op(L); end;

Mathematica

a[n_]:=(2-2^(1-n))/n; a[1]:=1; Table[Numerator[a[n]], {n, 40}]
a[n_]:=a[n-1]+(2^(1-n)*(1+n)-2)/((n-1)*n); a[1]:=1; Table[Numerator[a[n]], {n, 40}]
a[n_]:=a[n-1]*(2^n-1)*(n-1)/(n*(2^n-2)); a[1]:=1; Table[Numerator[a[n]], {n, 40}]

Formula

From Peter J. C. Moses, May 27 2004: (Start)

F(n) = (2-2^(1-n))/n.

G.f. for F: 2*(log(1-x/2)-log(1-x)).

E.g.f. for F: Integral 2*(e^x-e^(x/2))/x dx. (End)