A247685 Decimal expansion of the integral over the square (0,1)x(0,1) of 1/((x+y)*sqrt((1-x)*(1-y))) dx dy.

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

Offset

1,1

Comments

Also hyperbolic volume of the Whitehead link complement and (-2,3,8) pretzel link complement. This is the minimal volume attainable by a two-cusped orientable hyperbolic 3-manifold. - Jeremy Tan, Nov 17 2016

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Ian Agol, The minimal volume orientable hyperbolic 2-cusped 3-manifolds, Proc. Amer. Math. Soc., Vol. 138, No. 10 (2010), pp. 3723-3732. MR2661571

David H. Bailey and Jonathan M. Borwein, Highly Parallel, High-Precision Numerical Integration, Lawrence Berkeley National Laboratory, 2005, p. 9.

Eric Weisstein's World of Mathematics, Lerch Transcendent.

Wikipedia, Lerch zeta function.

Formula

Equals 4*Catalan.

Equals Integral_{x=0..Pi/2} log((1+cos(x))/(1-cos(x))) dx = Integral_{x=0..Pi/2} log((1+sin(x))/(1-sin(x))) dx. - Amiram Eldar, Apr 07 2022

From Amiram Eldar, Aug 14 2023: (Start)

Equals Phi(-1, 2, 1/2) = Sum_{k>=0} (-1)^k/(k+1/2)^2, where Phi is the Lerch transcendent.

Equals Integral_{x=-Pi/2..Pi/2} x/sin(x) dx. (End)

Example

3.663862376708876060218414059729536443096597497126688537...

Maple

evalf(4*Catalan, 130);  # Alois P. Heinz, Aug 14 2023

Mathematica

RealDigits[4*Catalan, 10, 103] // First

Prog

(PARI) default(realprecision, 100); 4*Catalan \\ G. C. Greubel, Aug 25 2018
(Magma) SetDefaultRealField(RealField(100)); R:=RealField(); 4*Catalan(R); // G. C. Greubel, Aug 25 2018

Crossrefs

Cf. A006752 (Catalan), A091518, A244921, A247674, A247675, A247677, A247684.

Sequence in context: A016662 A258271 A105396 * A185735 A339021 A197263

Adjacent sequences: A247682 A247683 A247684 * A247686 A247687 A247688

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Sep 22 2014

Status

approved