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
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
KEYWORD
nonn,cons
AUTHOR
Eric W. Weisstein, Jan 17 2004
STATUS
approved