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A152648
Decimal expansion of 2*zeta(3).
18
2, 4, 0, 4, 1, 1, 3, 8, 0, 6, 3, 1, 9, 1, 8, 8, 5, 7, 0, 7, 9, 9, 4, 7, 6, 3, 2, 3, 0, 2, 2, 8, 9, 9, 9, 8, 1, 5, 2, 9, 9, 7, 2, 5, 8, 4, 6, 8, 0, 9, 9, 7, 7, 6, 3, 5, 8, 4, 5, 4, 3, 1, 1, 0, 6, 8, 3, 6, 7, 6, 4, 1, 1, 5, 7, 2, 6, 2, 6, 1, 8, 0, 3, 7, 2, 9, 1, 1, 7, 4, 7, 2, 1, 8, 6, 7, 0, 5, 1, 6, 2, 9, 2, 3, 9
OFFSET
1,1
COMMENTS
A division by 2 is missing in Mezo's penultimate formula on page 4.
This constant is irrational but not known to be transcendental. - Charles R Greathouse IV, Sep 02 2024
REFERENCES
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 43.
LINKS
Ilham A. Aliev and Ayhan Dil, Tornheim-like series, harmonic numbers and zeta values, arXiv:2008.02488 [math.NT], 2020, p. 2.
R. Barbieri, J. A. Mignaco, and E. Remiddi, Electron form factors up to fourth order. I., Il Nuovo Cim. 11A (4) (1972) 824-864, Table II. (3).
David Borwein and J. M. Borwein, On an intriguing integral and some series related to zeta(4), Proc. Am. Math. Soc. 123 (1995) 1191-1198.
Istvan Mezo, Summation of Hyperharmonic Numbers, arXiv:0811.0042 [math.CO], 2008.
Michael Penn, a nice double sum., YouTube video, 2020.
FORMULA
Equals 2*A002117 = Sum_{j>=1} H(j)/j^2 where H(j) = A001008(j)/A002805(j).
Equals Integral_{x>=0} x^2/(exp(x)-1). - Jean-François Alcover, Nov 12 2013
Equals Sum_{m>=1} Sum_{n>=1} 1/(m*n*(m + n)). - Jean-François Alcover, Jun 17 2020
Equals Integral_{x=0..1} log(x)^2/(1-x) dx. - Amiram Eldar, Aug 03 2020
Equals the absolute value of psi''(1) = -2.404..., the 2nd derivative of the digamma function at 1. - R. J. Mathar, Aug 29 2023
EXAMPLE
Equals 2.4041138063191885707994...
MATHEMATICA
RealDigits[2*Zeta[3], 10, 120][[1]] (* Harvey P. Dale, Dec 02 2011 *)
PROG
(PARI) default(realprecision, 20080); x=2*zeta(3); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b152648.txt", n, " ", d)); \\ Harry J. Smith, Jul 12 2009
CROSSREFS
Cf. A060804 (continued fraction).
Sequence in context: A054003 A338475 A134352 * A327898 A140875 A364315
KEYWORD
cons,easy,nonn,changed
AUTHOR
R. J. Mathar, Dec 10 2008
STATUS
approved