A091518 Decimal expansion of the hyperbolic volume of the figure eight knot complement.
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, 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, 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
Offset: 1
Examples
2.02988321281930725004240510854904057188337861506059958403497821355319...
References
- 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.
Links
- David H. Bailey and Jonathan M. Borwein, Experimental Mathematics: Examples, Methods and Implications, Notices of the AMS, Vol. 52, No. 5 (2005), pp. 502-514. See p. 504.
- Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 638.
- John Milnor, Topology through the centuries: Low dimensional manifolds, Bull. Amer. Math. Soc., Vol. 52, No. 4 (2015), pp. 545-584; see p. 562.
- Eric Weisstein's World of Mathematics, Figure Eight Knot.
Programs
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Mathematica
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 *) N[(PolyGamma[1, 1/3] - PolyGamma[1, 2/3]) / (2*Sqrt[3]), 105] (* Vaclav Kotesovec, Jun 17 2021 *) -
PARI
2*suminf(k=0,binomial(2*k,k)/16^k/(2*k+1)^2) \\ Charles R Greathouse IV, Oct 15 2014
Formula
Equals -6 * Integral_{x=0..Pi/3} log|2*sin(x)| dx. - Jonathan Sondow, Oct 15 2015
From Amiram Eldar, Jul 07 2021: (Start)
Equals 2*sqrt(3) * Sum_{n>=1} ((1/(n*binomial(2*n,n))) * (Sum_{k=n..(2*n-1)} 1/k)).
Equals 2*Sum_{k>=0} binomial(2*k,k)/(16^k*(2*k+1)^2).
Equals 2*Sum_{k>=1} sin(k*Pi/3)/k^2. (End)
Equals polygamma(1, 1/3)/sqrt(3) - 2*Pi^2/3^(3/2). - Vaclav Kotesovec, Jul 07 2021