A152648 Decimal expansion of 2*zeta(3).

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 Mező'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

Harry J. Smith, Table of n, a(n) for n = 1..20000

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 Jonathan M. Borwein, On an intriguing integral and some series related to zeta(4), Proc. Am. Math. Soc. 123 (1995) 1191-1198.

J. W. Elder, The dispersion of marked fluid in turbulent shear flow, J. Fluid Mech. 5 (1959) 544-560. See equation 14.

István Mező, Summation of Hyperharmonic Series, arXiv:0811.0042 [math.CO], 2008.

Michael Penn, a nice double sum., YouTube video, 2020.

Michael Penn, Euler's harmonic number identity, 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

Equals 2 + Integral_{x=0..1} (1-x)*log(1-x)^2/x dx [Elder, 1959]. - Chris R. Rehmann, Apr 19 2026

Example

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. A002117, A001008, A002805.

Cf. A060804 (continued fraction).

Sequence in context: A054003 A338475 A134352 * A327898 A140875 A364315

Adjacent sequences: A152645 A152646 A152647 * A152649 A152650 A152651

Keyword

cons,easy,nonn

Author

R. J. Mathar, Dec 10 2008

Status

approved