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%I #34 Jul 07 2021 07:17:14
%S 2,0,2,9,8,8,3,2,1,2,8,1,9,3,0,7,2,5,0,0,4,2,4,0,5,1,0,8,5,4,9,0,4,0,
%T 5,7,1,8,8,3,3,7,8,6,1,5,0,6,0,5,9,9,5,8,4,0,3,4,9,7,8,2,1,3,5,5,3,1,
%U 9,4,9,5,2,5,1,6,4,8,8,0,4,4,2,7,2,9,4,0,7,0,8,4,5,6,5,1,3,3,8,9,8,9
%N Decimal expansion of the hyperbolic volume of the figure eight knot complement.
%D David H. Bailey, Jonathan M. Borwein, Neil J. Calkin, Roland Girgensohn, D. Russell Luke and Victor H. Moll, Experimental Mathematics in Action, Wellesley, MA: A K Peters, 2007, p. 38.
%H David H. Bailey and Jonathan M. Borwein, <a href="http://www.ams.org/notices/200505/fea-borwein.pdf">Experimental Mathematics: Examples, Methods and Implications</a>, Notices of the AMS, Vol. 52, No. 5 (2005), pp. 502-514. See p. 504.
%H Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 638.
%H John Milnor, <a href="http://dx.doi.org/10.1090/bull/1507">Topology through the centuries: Low dimensional manifolds</a>, Bull. Amer. Math. Soc., Vol. 52, No. 4 (2015), pp. 545-584; see p. 562.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FigureEightKnot.html">Figure Eight Knot</a>.
%F Equals -6 * Integral_{x=0..Pi/3} log|2*sin(x)| dx. - _Jonathan Sondow_, Oct 15 2015
%F From _Amiram Eldar_, Jul 07 2021: (Start)
%F Equals 2*sqrt(3) * Sum_{n>=1} ((1/(n*binomial(2*n,n))) * (Sum_{k=n..(2*n-1)} 1/k)).
%F Equals 2*Sum_{k>=0} binomial(2*k,k)/(16^k*(2*k+1)^2).
%F Equals 2*Sum_{k>=1} sin(k*Pi/3)/k^2. (End)
%F Equals polygamma(1, 1/3)/sqrt(3) - 2*Pi^2/3^(3/2). - _Vaclav Kotesovec_, Jul 07 2021
%e 2.02988321281930725004240510854904057188337861506059958403497821355319...
%t RealDigits[N[2*Pi/3 - 1/18*HypergeometricPFQ[{3/2, 3/2, 3/2}, {5/2, 5/2}, 1/4], 102]][[1]] (* _Jean-François Alcover_, Nov 12 2012, after _Eric W. Weisstein_ *)
%t N[(PolyGamma[1, 1/3] - PolyGamma[1, 2/3]) / (2*Sqrt[3]), 105] (* _Vaclav Kotesovec_, Jun 17 2021 *)
%o (PARI) 2*suminf(k=0,binomial(2*k,k)/16^k/(2*k+1)^2) \\ _Charles R Greathouse IV_, Oct 15 2014
%K nonn,cons
%O 1,1
%A _Eric W. Weisstein_, Jan 17 2004
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