OFFSET
1,2
REFERENCES
S. R. Finch, Mathematical Constants, Cambridge, 2003; see Section 3.10, Kneser-Mahler polynomial constants, p. 232, and Section 5.23, Monomer-dimer constants, p. 408.
LINKS
Jonathan M. Borwein, Armin Straub, James Wan, and Wadim Zudilin, Densities of Short Uniform Random Walks, Canad. J. Math. 64(1) (2012), 961-990; see p. 978.
Henry Cohn, Richard Kenyon, and James Propp, A variational principle for domino tilings, arXiv:math/0008220 [math.CO], 2000.
Henry Cohn, Richard Kenyon, and James Propp, A variational principle for domino tilings, J. Amer. Math. Soc. 14(2) (2000), 297-346.
Kurt Mahler, A remark on a paper of mine on polynomials. [In this paper, j is log(beta) = A244996.]
Kurt Mahler, A remark on a paper of mine on polynomials, Illinois J. Math. 8(1) (1964), 1-4.
Francisco Santos, The Cayley trick and triangulations of products of simplices, arXiv:math/0312069 [math.CO], 2004; see part (2) of Theorem 1 (p. 2, possible typo), Lemma 4.8 (p. 22), and Theorem 4.9 (p. 22).
Francisco Santos, The Cayley trick and triangulations of products of simplices, Cont. Math. 374 (2005), 151-177.
Eric Weisstein's MathWorld, Clausen's Integral.
Eric Weisstein's MathWorld, Gieseking's Constant.
Eric Weisstein's MathWorld, Lobachevsky's Function.
Wikipedia, Clausen function.
FORMULA
beta = exp(G/Pi) = exp((PolyGamma(1, 4/3) - PolyGamma(1, 2/3) + 9)/(4*sqrt(3)*Pi)), where G is Gieseking's constant (cf. A143298) and PolyGamma(1,z) the first derivative of the digamma function psi(z).
Also equals exp(-Im(Li_2( 1/2 - (i*sqrt(3))/2))/Pi), where Li_2 is the dilogarithm function.
EXAMPLE
1.38135644451849779337146695685...
MATHEMATICA
Exp[(PolyGamma[1, 4/3] - PolyGamma[1, 2/3] + 9)/(4*Sqrt[3]*Pi)] // RealDigits[#, 10, 100]& // First
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Jean-François Alcover, May 21 2014
STATUS
approved