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

1,4

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

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, p. 233.

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.

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

Example

1.0149416064096536250...

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

Prog

(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

Crossrefs

Cf. A091518, A242710, A244345, A244996.

Sequence in context: A178143 A070435 A070516 * A177839 A013669 A085365

Adjacent sequences: A143295 A143296 A143297 * A143299 A143300 A143301

Keyword

nonn,cons

Author

Eric W. Weisstein, Aug 05 2008

Status

approved