OFFSET
0,1
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..10000
D. Borwein, J. M. Borwein, M. L. Glasser, J. G Wan, Moments of Ramanujan's Generalized Elliptic Integrals and Extensions of Catalan's Constant, 2010.
D. Borwein, J. M. Borwein, M. L. Glasser, J. G Wan, Moments of Ramanujan's Generalized Elliptic Integrals and Extensions of Catalan's Constant, Journal of Mathematical Analysis and Applications, Volume 384, Issue 2, 15 December 2011, Pages 478-496.
FORMULA
G(s) = (Pi/4) * 3F2(1/2, 1/2-s, s+1/2; 1, 3/2; 1), with 2F1 the hypergeometric function.
G(s) = (1/(8*s))*(Pi + cos(Pi*s)*(psi(1/4+s/2) - psi(3/4+s/2))), where psi is the digamma function (PolyGamma).
G(1/3) = (3/8)*sqrt(3)*log(2 + sqrt(3)) = (3/4)*sqrt(3)*arccoth(sqrt(3)).
EXAMPLE
0.855389245838564640972481036740456552226831197315573480398142...
MATHEMATICA
RealDigits[(3/8)*Sqrt[3]*Log[2 + Sqrt[3]], 10, 107] // First
N[Pi*HypergeometricPFQ[{1/6, 1/2, 5/6}, {1, 3/2}, 1]/4, 105] (* Vaclav Kotesovec, Apr 24 2015 *)
PROG
(PARI) (3/8)*sqrt(3)*log(2 + sqrt(3)) \\ G. C. Greubel, Aug 24 2018
(Magma) SetDefaultRealField(RealField(100)); (3/8)*Sqrt(3)*Log(2 + Sqrt(3)); // G. C. Greubel, Aug 24 2018
CROSSREFS
KEYWORD
AUTHOR
Jean-François Alcover, Apr 23 2015
STATUS
approved