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%I #32 Jan 04 2024 06:37:11
%S 1,3,7,15,31,21,127,255,511,1023,2047,1365,8191,16383,32767,65535,
%T 131071,29127,524287,209715,299593,4194303,8388607,5592405,33554431,
%U 67108863,134217727,268435455,536870911,357913941,2147483647,4294967295,8589934591,17179869183
%N 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).
%C 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
%C Also, numerators of BINOMIAL transform of sequence [1, 1/2, 1/3, 1/4, ...]. - _Gary W. Adamson_, Apr 26 2005
%D Putnam Competition, 2003, Problem B2.
%H Alois P. Heinz, <a href="/A090633/b090633.txt">Table of n, a(n) for n = 1..3320</a> (first 200 terms from T. D. Noe)
%H F. Nedermeyer and Y. Smorodinsky, <a href="http://static.nsta.org/pdfs/QuantumV7N1.pdf">Resistance in the multidimensional cube</a>, Quantum, Sept/October 1996, pp. 12-15 (beware file is 75Mb).
%F From _Peter J. C. Moses_, May 27 2004: (Start)
%F F(n) = (2-2^(1-n))/n.
%F G.f. for F: 2*(log(1-x/2)-log(1-x)).
%F E.g.f. for F: Integral 2*(e^x-e^(x/2))/x dx. (End)
%e 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, ...
%p 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)];
%p F := proc(n) local L,i; L := f2(n); for i from 1 to n-1 do L := f(L); od: op(L); end;
%t a[n_]:=(2-2^(1-n))/n; a[1]:=1; Table[Numerator[a[n]], {n, 40}]
%t a[n_]:=a[n-1]+(2^(1-n)*(1+n)-2)/((n-1)*n); a[1]:=1; Table[Numerator[a[n]], {n, 40}]
%t a[n_]:=a[n-1]*(2^n-1)*(n-1)/(n*(2^n-2)); a[1]:=1; Table[Numerator[a[n]], {n, 40}]
%o (Haskell)
%o import Data.Ratio (numerator, (%))
%o a090633 n = numerator z where
%o [z] = (until ((== 1) . length) avg) $ map (1 %) [1..n]
%o avg xs = zipWith (\x x' -> (x + x') / 2) (tail xs) xs
%o -- _Reinhard Zumkeller_, Dec 08 2011
%Y Cf. A090634 (denominators).
%K nonn,frac
%O 1,2
%A _N. J. A. Sloane_, Dec 13 2003
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