A143298 - OEIS

%I #58 Dec 24 2024 09:37:54

%S 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,

%T 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,

%U 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

%N Decimal expansion of Gieseking's constant.

%C 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

%D J. Borwein and P. Borwein, Experimental and computational mathematics: Selected writings, Perfectly Scientific Press, 2010, p. 106.

%D Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 233, 512.

%H G. C. Greubel, <a href="/A143298/b143298.txt">Table of n, a(n) for n = 1..10000</a>

%H Colin C. Adams, <a href="http://www.jstor.org/stable/2690774">The newest inductee in the Number Hall of Fame</a>, Mathematics Magazine, Vol. 71, No. 5 (1998), pp. 341-349.

%H John Campbell, <a href="https://hal.archives-ouvertes.fr/hal-03644515">Proof of a conjecture due to Sun concerning Catalan's constant</a>, hal-03644515 [math], 2022.

%H John M. Campbell, <a href="https://hosted.math.rochester.edu/ojac/vol18/290.pdf">On two conjectures due to Sun</a>, Univ. Rochester, Online J. Analytic Comb. (2023) Issue 18, Art No. 4. See p. 4.

%H P. J. de Doelder, <a href="http://dx.doi.org/10.1016/0377-0427(84)90007-4">On the Clausen integral Cl_2(theta) and a related integral</a>, J. Comp. Appl. Math. 11 (1984) 325-330.

%H K. S. Kolbig, <a href="http://dx.doi.org/10.1016/0377-0427(95)00150-6">Chebyshev coefficients for the Clausen function Cl_2(x)</a>, J. Comp. Appl. Math. 64 (1995) 295-297.

%H Vincent Nguyen, <a href="https://arxiv.org/abs/2304.11614">On Some Series Involving Harmonic and Skew-Harmonic Numbers</a>, arXiv:2304.11614 [math.CA], 2023.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GiesekingsConstant.html">Gieseking's Constant</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Gieseking_manifold">Gieseking manifold</a>.

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

%F 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.

%F From _Amiram Eldar_, Aug 14 2020: (Start)

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

%F 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.

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

%F Equals A091518/2 = A244345/5. - _Hugo Pfoertner_, Sep 16 2024

%e 1.0149416064096536250...

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

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

%o (PARI)

%o polygamma(n, x) = if (n == 0, psi(x), (-1)^(n+1)*n!*zetahurwitz(n+1, x));

%o sqrt(3)/6*(polygamma(1, 1/3) - 2*Pi^2/3)

%o (9 - polygamma(1, 2/3) + polygamma(1, 4/3))/(4*sqrt(3)) \\ _Gheorghe Coserea_, Sep 30 2018

%o (PARI)

%o clausen(n, x) = my(z = polylog(n, exp(I*x))); if (n%2, real(z), imag(z));

%o clausen(2, Pi/3) \\ _Gheorghe Coserea_, Sep 30 2018

%o (PARI)

%o 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

%Y Cf. A091518, A242710, A244345, A244996.

%K nonn,cons

%O 1,4

%A _Eric W. Weisstein_, Aug 05 2008