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%I #18 Apr 05 2024 11:10:14
%S 8,5,5,3,8,9,2,4,5,8,3,8,5,6,4,6,4,0,9,7,2,4,8,1,0,3,6,7,4,0,4,5,6,5,
%T 5,2,2,2,6,8,3,1,1,9,7,3,1,5,5,7,3,4,8,0,3,9,8,1,4,2,0,0,4,0,4,2,5,6,
%U 2,0,1,2,9,8,6,7,7,4,5,9,7,1,5,7,0,1,5,6,6,0,3,9,8,2,9,8,2,6,5,0,5,4,6,6,6,7,5
%N Decimal expansion of G(1/3), a generalized Catalan constant.
%H G. C. Greubel, <a href="/A257436/b257436.txt">Table of n, a(n) for n = 0..10000</a>
%H D. Borwein, J. M. Borwein, M. L. Glasser, J. G Wan, <a href="https://carmamaths.org/resources/jon/emoments.pdf">Moments of Ramanujan's Generalized Elliptic Integrals and Extensions of Catalan's Constant</a>, 2010.
%H D. Borwein, J. M. Borwein, M. L. Glasser, J. G Wan, <a href="https://doi.org/10.1016/j.jmaa.2011.06.001">Moments of Ramanujan's Generalized Elliptic Integrals and Extensions of Catalan's Constant</a>, Journal of Mathematical Analysis and Applications, Volume 384, Issue 2, 15 December 2011, Pages 478-496.
%F G(s) = (Pi/4) * 3F2(1/2, 1/2-s, s+1/2; 1, 3/2; 1), with 2F1 the hypergeometric function.
%F 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).
%F G(1/3) = (3/8)*sqrt(3)*log(2 + sqrt(3)) = (3/4)*sqrt(3)*arccoth(sqrt(3)).
%e 0.855389245838564640972481036740456552226831197315573480398142...
%t RealDigits[(3/8)*Sqrt[3]*Log[2 + Sqrt[3]], 10, 107] // First
%t N[Pi*HypergeometricPFQ[{1/6, 1/2, 5/6}, {1, 3/2}, 1]/4, 105] (* _Vaclav Kotesovec_, Apr 24 2015 *)
%o (PARI) (3/8)*sqrt(3)*log(2 + sqrt(3)) \\ _G. C. Greubel_, Aug 24 2018
%o (Magma) SetDefaultRealField(RealField(100)); (3/8)*Sqrt(3)*Log(2 + Sqrt(3)); // _G. C. Greubel_, Aug 24 2018
%Y Cf. A006752 (G(0) = Catalan), A257435 (G(1/6)), A091648 (G(1/4)), A257437 (G(1/12)), A257438 (G(1/5)).
%K nonn,cons,easy
%O 0,1
%A _Jean-François Alcover_, Apr 23 2015
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