OFFSET
1,1
COMMENTS
Also hyperbolic volume of the Whitehead link complement and (-2,3,8) pretzel link complement. This is the minimal volume attainable by a two-cusped orientable hyperbolic 3-manifold. - Jeremy Tan, Nov 17 2016
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..10000
Ian Agol, The minimal volume orientable hyperbolic 2-cusped 3-manifolds, Proc. Amer. Math. Soc., Vol. 138, No. 10 (2010), pp. 3723-3732. MR2661571
David H. Bailey and Jonathan M. Borwein, Highly Parallel, High-Precision Numerical Integration, Lawrence Berkeley National Laboratory, 2005, p. 9.
Eric Weisstein's World of Mathematics, Lerch Transcendent.
Wikipedia, Lerch zeta function.
FORMULA
Equals 4*Catalan.
Equals Integral_{x=0..Pi/2} log((1+cos(x))/(1-cos(x))) dx = Integral_{x=0..Pi/2} log((1+sin(x))/(1-sin(x))) dx. - Amiram Eldar, Apr 07 2022
From Amiram Eldar, Aug 14 2023: (Start)
Equals Phi(-1, 2, 1/2) = Sum_{k>=0} (-1)^k/(k+1/2)^2, where Phi is the Lerch transcendent.
Equals Integral_{x=-Pi/2..Pi/2} x/sin(x) dx. (End)
EXAMPLE
3.663862376708876060218414059729536443096597497126688537...
MAPLE
evalf(4*Catalan, 130); # Alois P. Heinz, Aug 14 2023
MATHEMATICA
RealDigits[4*Catalan, 10, 103] // First
PROG
(PARI) default(realprecision, 100); 4*Catalan \\ G. C. Greubel, Aug 25 2018
(Magma) SetDefaultRealField(RealField(100)); R:=RealField(); 4*Catalan(R); // G. C. Greubel, Aug 25 2018
CROSSREFS
KEYWORD
AUTHOR
Jean-François Alcover, Sep 22 2014
STATUS
approved