A143298 Decimal expansion of Gieseking's constant.

1, 0, 1, 4, 9, 4, 1, 6, 0, 6, 4, 0, 9, 6, 5, 3, 6, 2, 5, 0, 2, 1, 2, 0, 2, 5, 5, 4, 2, 7, 4, 5, 2, 0, 2, 8, 5, 9, 4, 1, 6, 8, 9, 3, 0, 7, 5, 3, 0, 2, 9, 9, 7, 9, 2, 0, 1, 7, 4, 8, 9, 1, 0, 6, 7, 7, 6, 5, 9, 7, 4, 7, 6, 2, 5, 8, 2, 4, 4, 0, 2, 2, 1, 3, 6, 4, 7, 0, 3, 5, 4, 2, 2, 8, 2, 5, 6, 6, 9, 4, 9, 4, 5, 8, 6

Offset: 1

Author

Eric W. Weisstein, Aug 05 2008

Keywords

• nonn
• cons

Comments

The largest possible volume of a tetrahedron in hyperbolic space. Named by Adams (1998) after German mathematician Hugo Gieseking (1887 - 1915). - Amiram Eldar, Aug 14 2020

Examples

			1.0149416064096536250...
		

References

• J. Borwein and P. Borwein, Experimental and computational mathematics: Selected writings, Perfectly Scientific Press, 2010, p. 106.
• Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 233, 512.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Colin C. Adams, The newest inductee in the Number Hall of Fame, Mathematics Magazine, Vol. 71, No. 5 (1998), pp. 341-349.
• John Campbell, Proof of a conjecture due to Sun concerning Catalan's constant, hal-03644515 [math], 2022.
• John M. Campbell, On two conjectures due to Sun, Univ. Rochester, Online J. Analytic Comb. (2023) Issue 18, Art No. 4. See p. 4.
• P. J. de Doelder, On the Clausen integral Cl_2(theta) and a related integral, J. Comp. Appl. Math. 11 (1984) 325-330.
• K. S. Kolbig, Chebyshev coefficients for the Clausen function Cl_2(x), J. Comp. Appl. Math. 64 (1995) 295-297.
• Vincent Nguyen, On Some Series Involving Harmonic and Skew-Harmonic Numbers, arXiv:2304.11614 [math.CA], 2023.
• Eric Weisstein's World of Mathematics, Gieseking's Constant.
• Wikipedia, Gieseking manifold.

Crossrefs

Cf. A091518, A242710, A244345, A244996.

Programs

Maple

sqrt(3)/6*(Psi(1,1/3)-2*Pi^2/3) ; evalf(%) ; # R. J. Mathar, Sep 23 2013

Mathematica

N[(9 - PolyGamma[1, 2/3] + PolyGamma[1, 4/3])/(4*Sqrt[3]), 105] // RealDigits // First

PARI

polygamma(n, x) = if (n == 0, psi(x), (-1)^(n+1)*n!*zetahurwitz(n+1, x));
sqrt(3)/6*(polygamma(1, 1/3) - 2*Pi^2/3)
(9 - polygamma(1, 2/3) + polygamma(1, 4/3))/(4*sqrt(3)) \\ Gheorghe Coserea, Sep 30 2018

PARI

clausen(n, x) = my(z = polylog(n, exp(I*x))); if (n%2, real(z), imag(z));
clausen(2, Pi/3) \\ Gheorghe Coserea, Sep 30 2018

PARI

sqrt(3)/2 * sumpos(n=1, 1/(6*n-4)^2 + 1/(6*n-5)^2 - 1/(6*n-1)^2 - 1/(6*n-2)^2) \\ Gheorghe Coserea, Sep 30 2018

Formula

Equals (9 - PolyGamma(1, 2/3) + PolyGamma(1, 4/3))/(4*sqrt(3)).

Equals Sum_{k>0} sin(k*Pi/3)/k^2; (also equals (sqrt(3)/2)*Sum_{k>=1} -1/(6k-1)^2 - 1/(6k-2)^2 + 1/(6k-4)^2 + 1/(6k-5)^2). - Jean-François Alcover, Jun 19 2016, from the book by J. & P. Borwein.

From Amiram Eldar, Aug 14 2020: (Start)

Equals Integral_{x=0..2*Pi/3} log(2*cos(x/2)).

Equals (3*sqrt(3)/4) * (1 - Sum_{k>=0} 1/(3*k + 2)^2 + Sum_{k>=1} 1/(3*k + 1)^2) = (3*sqrt(3)/4) * Sum_{k>=1} A049347(k-1)/k^2.

Equals Pi * A244996 = Pi * log(A242710). (End)

Equals A091518/2 = A244345/5. - Hugo Pfoertner, Sep 16 2024