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A143298
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Decimal expansion of Gieseking's constant.
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22
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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
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OFFSET
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1,4
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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FORMULA
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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.
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.
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EXAMPLE
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1.0149416064096536250...
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MAPLE
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sqrt(3)/6*(Psi(1, 1/3)-2*Pi^2/3) ; evalf(%) ; # R. J. Mathar, Sep 23 2013
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MATHEMATICA
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N[(9 - PolyGamma[1, 2/3] + PolyGamma[1, 4/3])/(4*Sqrt[3]), 105] // RealDigits // First
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PROG
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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
(PARI)
clausen(n, x) = my(z = polylog(n, exp(I*x))); if (n%2, real(z), imag(z));
(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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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