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

1,2

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

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).

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)

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

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}]

Prog

(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

Crossrefs

Cf. A090634 (denominators).

Sequence in context: A336685 A069112 A064084 * A320024 A098583 A275531

Adjacent sequences: A090630 A090631 A090632 * A090634 A090635 A090636

Keyword

nonn,frac

Author

N. J. A. Sloane, Dec 13 2003

Status

approved