The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A091518

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A091518

Decimal expansion of the hyperbolic volume of the figure eight knot complement.
(history; published version)

#34 by Joerg Arndt at Wed Jul 07 07:17:14 EDT 2021

STATUS

reviewed

approved

#33 by Vaclav Kotesovec at Wed Jul 07 05:08:39 EDT 2021

STATUS

proposed

reviewed

#32 by Vaclav Kotesovec at Wed Jul 07 05:08:34 EDT 2021

STATUS

editing

proposed

#31 by Vaclav Kotesovec at Wed Jul 07 05:08:19 EDT 2021

FORMULA

Equals polygamma(1, 1/3)/sqrt(3) - 2*Pi^2/3^(3/2). - Vaclav Kotesovec, Jul 07 2021

Discussion

Wed Jul 07

05:08

Vaclav Kotesovec: One more formula.

#30 by Vaclav Kotesovec at Wed Jul 07 04:54:34 EDT 2021

EXAMPLE

2.0298832102988321281930725004240510854904057188337861506059958403497821355319...

STATUS

proposed

editing

#29 by Amiram Eldar at Wed Jul 07 02:45:36 EDT 2021

STATUS

editing

proposed

#28 by Amiram Eldar at Wed Jul 07 02:31:11 EDT 2021

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

D. David H. Bailey and J. Jonathan M. Borwein, <a href="http://www.ams.org/notices/200505/fea-borwein.pdf">Experimental Mathematics: Examples, Methods and Implications</a> p, Notices of the AMS, Vol. 52, No. 4-5 (2005), pp. 502-514. See p. 504.

J. 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.

Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FigureEightKnot.html">Figure Eight Knot</a>.

FORMULA

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)

#27 by Amiram Eldar at Wed Jul 07 02:27:52 EDT 2021

COMMENTS

Decimal expansion of -6*int_{x=0..Pi/3} log|2*sin(x)| dx. - Jonathan Sondow, Oct 15 2015

FORMULA

Equals -6 * Integral_{x=0..Pi/3} log|2*sin(x)| dx. - Jonathan Sondow, Oct 15 2015

STATUS

approved

editing

#26 by Vaclav Kotesovec at Thu Jun 17 09:25:40 EDT 2021

STATUS

editing

approved

#25 by Vaclav Kotesovec at Thu Jun 17 09:25:35 EDT 2021

LINKS

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.

STATUS

approved

editing