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

1,1

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

Table of n, a(n) for n=1..102.

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.

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

Example

2.02988321281930725004240510854904057188337861506059958403497821355319...

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 *)

Prog

(PARI) 2*suminf(k=0, binomial(2*k, k)/16^k/(2*k+1)^2) \\ Charles R Greathouse IV, Oct 15 2014

Crossrefs

Sequence in context: A324305 A265607 A332628 * A096734 A220234 A038020

Adjacent sequences: A091515 A091516 A091517 * A091519 A091520 A091521

Keyword

nonn,cons

Author

Eric W. Weisstein, Jan 17 2004

Status

approved