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Denominators in the coefficients that form the odd-indexed partial quotients of the continued fraction representation of the inverse tangent of 1/x.
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%I #15 Aug 11 2014 22:45:27

%S 1,4,64,256,16384,65536,1048576,4194304,1073741824,4294967296,

%T 68719476736,274877906944,17592186044416,70368744177664,

%U 1125899906842624,4503599627370496,4611686018427387904

%N Denominators in the coefficients that form the odd-indexed partial quotients of the continued fraction representation of the inverse tangent of 1/x.

%C Limit A110257(n)/a(n) = limit A110255(2*n-1)/A110256(2*n-1) = 4/Pi.

%C Apart from offset, identical to A056982.

%H Paul D. Hanna, <a href="/A110258/b110258.txt">Table of n, a(n) for n = 1..200</a>

%H <a href="/index/Di#divseq">Index to divisibility sequences</a>

%F a(n) = 4^A005187(n-1).

%F a(n) = A110256(2*n-1).

%e arctan(1/x) = 1/x - 1/(3*x^3) + 1/(5*x^5) - 1/(7*x^7) +-...

%e = [0; x, 3*x, 5/4*x, 28/9*x, 81/64*x, 704/225*x, 325/256*x,

%e 768/245*x, 20825/16384*x, 311296/99225*x, 83349/65536*x,

%e 1507328/480249*x, 1334025/1048576*x, 3145728/1002001*x,...]

%e = 1/(x + 1/(3*x + 1/(5/4*x + 1/(28/9*x + 1/(81/64*x +...))))).

%e The coefficients of x in the even-indexed partial quotients converge to Pi:

%e {3, 28/9, 704/225, 768/245, 311296/99225, ...}.

%e The coefficients of x in the odd-indexed partial quotients converge to 4/Pi:

%e {1, 5/4, 81/64, 325/256, 20825/16384, ...}.

%o (PARI) {a(n)=denominator(subst((contfrac( sum(k=0,2*n+1,(-1)^k/x^(2*k+1)/(2*k+1)),2*n+2))[2*n],x,1))}

%o (PARI) a(n)=4^(2*n-vecsum(binary(n-1))-2) \\ _Charles R Greathouse IV_, Apr 09 2012

%Y See A056982 for another version of this sequence.

%Y Cf. A110257 (numerators), A110255/A110256 (continued fraction), A110259/A110260.

%K frac,nonn,easy

%O 1,2

%A _Paul D. Hanna_, Jul 18 2005

%E Edited by _N. J. A. Sloane_, Jun 05 2007