A257436 Decimal expansion of G(1/3), a generalized Catalan constant.

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, 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, 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

Offset: 0

Author

Jean-François Alcover, Apr 23 2015

Keywords

• nonn
• cons
• easy

Examples

			0.855389245838564640972481036740456552226831197315573480398142...
		

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.

Crossrefs

Cf. A006752 (G(0) = Catalan), A257435 (G(1/6)), A091648 (G(1/4)), A257437 (G(1/12)), A257438 (G(1/5)).

Programs

Magma

SetDefaultRealField(RealField(100)); (3/8)*Sqrt(3)*Log(2 + Sqrt(3)); // G. C. Greubel, Aug 24 2018

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 *)

PARI

(3/8)*sqrt(3)*log(2 + sqrt(3)) \\ G. C. Greubel, Aug 24 2018

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)).