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
KEYWORD
nonn,frac
AUTHOR
N. J. A. Sloane, Dec 13 2003
STATUS
approved