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A257436
Decimal expansion of G(1/3), a generalized Catalan constant.
4
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
0,1
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(1/3) = (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
Cf. A006752 (G(0) = Catalan), A257435 (G(1/6)), A091648 (G(1/4)), A257437 (G(1/12)), A257438 (G(1/5)).
Sequence in context: A072991 A157414 A021543 * A201295 A011107 A356805
KEYWORD
nonn,cons,easy
AUTHOR
STATUS
approved