login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A091518 Decimal expansion of the hyperbolic volume of the figure eight knot complement. 1
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 (list; constant; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 23 23:07 EDT 2022. Contains 353993 sequences. (Running on oeis4.)