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
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.
Programs
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Maple
sqrt(3)/6*(Psi(1,1/3)-2*Pi^2/3) ; evalf(%) ; # R. J. Mathar, Sep 23 2013
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Mathematica
N[(9 - PolyGamma[1, 2/3] + PolyGamma[1, 4/3])/(4*Sqrt[3]), 105] // RealDigits // First
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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
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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
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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.
Comments