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A110255 Numerators in the fractional coefficients that form the partial quotients of the continued fraction representation of the inverse tangent of 1/x.

%I A110255 #14 Nov 12 2012 23:39:17
%S A110255 1,3,5,28,81,704,325,768,20825,311296,83349,1507328,1334025,3145728,
%T A110255 5337189,130023424,1366504425,7516192768,5466528925,12884901888,
%U A110255 87470372561,2954937499648,349899121845,12919261626368,22394407746529
%N A110255 Numerators in the fractional coefficients that form the partial quotients of the continued fraction representation of the inverse tangent of 1/x.
%C A110255 Limit a(2*n-1)/A110256(2*n-1) = limit A110257(n)/A110258(n) = 4/Pi.
%C A110255 Limit a(2*n)/A110256(2*n) = limit A110259(n)/A110260(n) = Pi.
%H A110255 Paul D. Hanna, <a href="/A110255/b110255.txt">Table of n, a(n) for n = 1..400</a>
%e A110255 arctan(1/x) = 1/x - 1/(3*x^3) + 1/(5*x^5) - 1/(7*x^7) +-...
%e A110255 = [0; x, 3*x, 5/4*x, 28/9*x, 81/64*x, 704/225*x, 325/256*x,
%e A110255 768/245*x, 20825/16384*x, 311296/99225*x, 83349/65536*x,
%e A110255 1507328/480249*x, 1334025/1048576*x, 3145728/1002001*x,...]
%e A110255 = 1/(x + 1/(3*x + 1/(5/4*x + 1/(28/9*x + 1/(81/64*x +...))))).
%e A110255 The coefficients of x in the even-indexed partial quotients converge to Pi:
%e A110255 {3, 28/9, 704/225, 768/245, 311296/99225, ...}.
%e A110255 The coefficients of x in the odd-indexed partial quotients converge to 4/Pi:
%e A110255 {1, 5/4, 81/64, 325/256, 20825/16384, ...}.
%o A110255 (PARI) {a(n)=numerator(subst((contfrac( sum(k=0,n,(-1)^k/x^(2*k+1)/(2*k+1)),n+1))[n+1],x,1))}
%Y A110255 Cf. A110256 (denominators), A110257/A110258 (odd-indexed), A110259/A110260 (even-indexed).
%K A110255 cofr,frac,nonn
%O A110255 1,2
%A A110255 _Paul D. Hanna_, Jul 18 2005