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A110260 Denominators in the coefficients that form the even-indexed partial quotients of the continued fraction representation of the inverse tangent of 1/x.

%I A110260 #12 Nov 12 2012 23:41:06
%S A110260 1,9,225,245,99225,480249,1002001,41409225,2393453205,4102737925,
%T A110260 940839860961,4113258565689,16802526820625,246430431820125,
%U A110260 21147754404155625,10036045423404225,98363281194784809225
%N A110260 Denominators in the coefficients that form the even-indexed partial quotients of the continued fraction representation of the inverse tangent of 1/x.
%C A110260 Limit A110259(n)/a(n) = limit A110255(2*n)/A110256(2*n) = Pi.
%H A110260 Paul D. Hanna, <a href="/A110260/b110260.txt">Table of n, a(n) for n = 1..200</a>
%F A110260 a(n) = A110256(2*n).
%e A110260 arctan(1/x) = 1/x - 1/(3*x^3) + 1/(5*x^5) - 1/(7*x^7) +-...
%e A110260 = [0; x, 3*x, 5/4*x, 28/9*x, 81/64*x, 704/225*x, 325/256*x,
%e A110260 768/245*x, 20825/16384*x, 311296/99225*x, 83349/65536*x,
%e A110260 1507328/480249*x, 1334025/1048576*x, 3145728/1002001*x,...]
%e A110260 = 1/(x + 1/(3*x + 1/(5/4*x + 1/(28/9*x + 1/(81/64*x +...))))).
%e A110260 The coefficients of x in the even-indexed partial quotients converge to Pi:
%e A110260 {3, 28/9, 704/225, 768/245, 311296/99225, ...}.
%e A110260 The coefficients of x in the odd-indexed partial quotients converge to 4/Pi:
%e A110260 {1, 5/4, 81/64, 325/256, 20825/16384, ...}.
%o A110260 (PARI) {a(n)=denominator(subst((contfrac( sum(k=0,2*n+2,(-1)^k/x^(2*k+1)/(2*k+1)),2*n+2))[2*n+1],x,1))}
%Y A110260 Cf. A110259 (numerators), A110255/A110256 (continued fraction), A110257/A110258.
%K A110260 frac,nonn
%O A110260 1,2
%A A110260 _Paul D. Hanna_, Jul 18 2005