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

Author

R. J. Mathar, Dec 10 2008

Keywords

• cons
• easy
• nonn

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

Examples

			Equals 2.4041138063191885707994...
		

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 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.
• Michael Penn, Euler's harmonic number identity, YouTube video, 2020.

Crossrefs

Cf. A002117, A001008, A002805.

Cf. A060804 (continued fraction).

Programs

Mathematica

RealDigits[2*Zeta[3],10,120][[1]] (* Harvey P. Dale, Dec 02 2011 *)

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

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