A257438 Decimal expansion of G(1/5), a generalized Catalan constant.

8, 9, 3, 6, 7, 1, 4, 2, 3, 4, 6, 0, 9, 6, 3, 5, 5, 4, 3, 0, 2, 0, 6, 9, 8, 5, 4, 5, 8, 3, 5, 4, 6, 0, 0, 7, 5, 4, 7, 5, 5, 8, 0, 9, 4, 7, 9, 6, 3, 2, 8, 0, 7, 8, 2, 2, 0, 3, 0, 8, 5, 8, 4, 8, 7, 8, 1, 5, 7, 6, 4, 1, 7, 7, 0, 4, 9, 2, 9, 1, 5, 0, 7, 9, 6, 7, 0, 5, 1, 6, 3, 8, 4, 2, 2, 3, 7, 2, 8, 1, 4, 8, 0, 3

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(1/5) = (5/8)*sqrt(5+2*sqrt(5))*(((sqrt(5)-1)/2)*arcsinh(sqrt(5+2*sqrt(5))) - arcsinh(sqrt(5-2*sqrt(5)))).

Example

0.8936714234609635543020698545835460075475580947963280782203...

Mathematica

RealDigits[(5/8)*Sqrt[5+2*Sqrt[5]]*(((Sqrt[5]-1)/2)*ArcSinh[Sqrt[5+2*Sqrt[5]]] - ArcSinh[Sqrt[5-2*Sqrt[5]]]), 10, 104] // First
N[Pi*HypergeometricPFQ[{3/10, 1/2, 7/10}, {1, 3/2}, 1]/4, 105] (* Vaclav Kotesovec, Apr 24 2015 *)

Prog

(PARI) (5/8)*sqrt(5+2*sqrt(5))*(((sqrt(5)-1)/2)*asinh(sqrt(5 +2*sqrt(5))) - asinh(sqrt(5-2*sqrt(5)))) \\ G. C. Greubel, Aug 24 2018
(Magma) SetDefaultRealField(RealField(100)); (5/8)*Sqrt(5+2*Sqrt(5))*(((Sqrt(5)-1)/2)*Argsinh(Sqrt(5+2*Sqrt(5))) - Argsinh(Sqrt(5-2*Sqrt(5)))); // G. C. Greubel, Aug 24 2018

Crossrefs

Cf. A006752 (G(0) = Catalan), A257435 (G(1/6)), A091648 (G(1/4)), A257436 (G(1/3)), A257437 (G(1/12)).

Sequence in context: A197392 A021922 A246843 * A197826 A197755 A360148

Adjacent sequences: A257435 A257436 A257437 * A257439 A257440 A257441

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Apr 23 2015

Status

approved