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A110259 Numerators in the coefficients that form the even-indexed partial quotients of the continued fraction representation of the inverse tangent of 1/x. 7

%I

%S 3,28,704,768,311296,1507328,3145728,130023424,7516192768,12884901888,

%T 2954937499648,12919261626368,52776558133248,774056185954304,

%U 66428094503714816,31525197391593472,308982963234634989568

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

%C Limit a(n)/A110260(n) = limit A110255(2*n)/A110256(2*n) = Pi.

%H Paul D. Hanna, <a href="/A110259/b110259.txt">Table of n, a(n) for n = 1..200</a>

%F a(n) = A110255(2*n).

%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)=numerator(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 Cf. A110260 (denominators), A110255/A110256 (continued fraction), A110257/A110258.

%K frac,nonn

%O 1,1

%A _Paul D. Hanna_, Jul 18 2005

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Last modified August 26 05:42 EDT 2019. Contains 326329 sequences. (Running on oeis4.)