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A257438
Decimal expansion of G(1/5), a generalized Catalan constant.
4
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
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
KEYWORD
nonn,cons,easy
AUTHOR
STATUS
approved