login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation to keep the OEIS running. In 2018 we replaced the server with a faster one, added 20000 new sequences, and reached 7000 citations (often saying "discovered thanks to the OEIS").
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A143298 Decimal expansion of Gieseking's constant. 20
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,4

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

C. C. Adams, The newest inductee in the Number Hall of Fame, Math. Mag., 71 (1998), 341-349.

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.

Eric Weisstein's World of Mathematics, Gieseking's Constant

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-Fran├žois Alcover, Jun 19 2016, from the book by J. & P. Borwein.

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

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 18 08:16 EST 2018. Contains 318219 sequences. (Running on oeis4.)