A152648 - OEIS

%I #50 Nov 01 2024 23:45:40

%S 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,

%T 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,

%U 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

%N Decimal expansion of 2*zeta(3).

%C A division by 2 is missing in Mezo's penultimate formula on page 4.

%C This constant is irrational but not known to be transcendental. - _Charles R Greathouse IV_, Sep 02 2024

%D Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 43.

%H Harry J. Smith, <a href="/A152648/b152648.txt">Table of n, a(n) for n = 1..20000</a>

%H Ilham A. Aliev and Ayhan Dil, <a href="https://arxiv.org/abs/2008.02488">Tornheim-like series, harmonic numbers and zeta values</a>, arXiv:2008.02488 [math.NT], 2020, p. 2.

%H R. Barbieri, J. A. Mignaco, and E. Remiddi, <a href="https://dx.doi.org/10.1007/BF02728545">Electron form factors up to fourth order. I.</a>, Il Nuovo Cim. 11A (4) (1972) 824-864, Table II. (3).

%H David Borwein and J. M. Borwein, <a href="https://doi.org/10.1090/S0002-9939-1995-1231029-X">On an intriguing integral and some series related to zeta(4)</a>, Proc. Am. Math. Soc. 123 (1995) 1191-1198.

%H Istvan Mezo, <a href="http://arxiv.org/abs/0811.0042">Summation of Hyperharmonic Numbers</a>, arXiv:0811.0042 [math.CO], 2008.

%H Michael Penn, <a href="https://www.youtube.com/watch?v=5KpGSMyUANU">a nice double sum.</a>, YouTube video, 2020.

%F Equals 2*A002117 = Sum_{j>=1} H(j)/j^2 where H(j) = A001008(j)/A002805(j).

%F Equals Integral_{x>=0} x^2/(exp(x)-1). - _Jean-François Alcover_, Nov 12 2013

%F Equals Sum_{m>=1} Sum_{n>=1} 1/(m*n*(m + n)). - _Jean-François Alcover_, Jun 17 2020

%F Equals Integral_{x=0..1} log(x)^2/(1-x) dx. - _Amiram Eldar_, Aug 03 2020

%F Equals the absolute value of psi''(1) = -2.404..., the 2nd derivative of the digamma function at 1. - _R. J. Mathar_, Aug 29 2023

%e Equals 2.4041138063191885707994...

%t RealDigits[2*Zeta[3],10,120][[1]] (* _Harvey P. Dale_, Dec 02 2011 *)

%o (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

%Y Cf. A002117, A001008, A002805.

%Y Cf. A060804 (continued fraction).

%K cons,easy,nonn,changed

%O 1,1

%A _R. J. Mathar_, Dec 10 2008