OFFSET
1,3
COMMENTS
The Dirichlet L-series for the principal character mod 6 (which is A120325 shifted left) evaluated at 2. - R. J. Mathar, Jul 20 2012
Equals the asymptotic mean of the abundancy index of the numbers coprime to 6 (A007310). - Amiram Eldar, May 12 2023
REFERENCES
F. Aubonnet, D. Guinin, and B.Joppin, Précis de Mathématiques, Analyse 2, Classes Préparatoires, Premier Cycle Universitaire, Bréal, 1990, Exercice 908, pages 82 and 91-92.
L. B. W. Jolley, Summation of Series, Dover, 1961.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..5000
Jean-Paul Allouche and Jeffrey Shallit, Sums of digits and the Hurwitz zeta function, in: K. Nagasaka and E. Fouvry (eds.), Analytic Number Theory, Lecture Notes in Mathematics, Vol. 1434, Springer, Berlin, Heidelberg, 1990, pp. 19-30.
Eugène-Charles Catalan, Mémoire sur la transformation des séries et sur quelques intégrales définies, Mémoires de l'Académie royale de Belgique, 1867, Vol. 33, pp. 1-50.
R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010-2015, Table 22.
Eric Weisstein's World of Mathematics, Digit Sum.
FORMULA
Equals 1 + (1/2)*(1/3)*(1/2) + (1/3)*(1*2)/(3*5)*(1/2)^2 + (1/4) *(1*2*3)/(3*5*7)*(1/2)^3 + .... [Jolley eq 277]
Equals 1/1^2 + 1/5^2 + 1/7^2 + 1/11^2 + 1/13^2 + 1/17^2 + .... - R. J. Mathar, Jul 20 2012
Equals 2*Sum_{n>=1} 1/(6*n*(3*n + (-1)^n - 3) - 3*(-1)^n + 5) = 2*Sum_{n>=1} 1/(2*A104777(n)). - Alexander R. Povolotsky, May 18 2014
Equals A019670^2. - Michel Marcus, May 19 2014
Equals 2*A086463 = 2*Sum_{n>=1} 1/A091999(n)^2, equivalent to the formula of 2012 above. - Alexander R. Povolotsky, May 20 2014
Equals 3F2(1,1,1; 3/2,2 ; 1/4), following from Clausen's formula of J. Reine Angew. Math 3 (1828) for squares of 2F1() as noted in A019670. - R. J. Mathar, Oct 16 2015
Equals Product_{n >= 3} prime(n)^2 / (prime(n)^2 - 1), Euler's prime product, excluding first two primes. - Fred Daniel Kline, Jun 09 2016
Equals Integral_{x=0..oo} log(x)/(x^6 - 1) dx. - Amiram Eldar, Aug 12 2020
Equals Sum_{k>=1} A000120(k) * (2*k+1)/(k^2*(k+1)^2) (Allouche and Shallit, 1990). - Amiram Eldar, Jun 01 2021
Equals Integral_{x=0..1} log(1+x+x^2)/x dx (Aubonnet). - Bernard Schott, Feb 04 2022
Equals Sum_{k>=1} A008833(k)/k^4. - Amiram Eldar, Jan 25 2024
Continued fraction expansion: 1/(1 - 1/(13 - 48/(34 - 270/(65 - ... - 2*(2*n-1)*n^3/((5*n^2+6*n+2) - ... ))))). See A130549. - Peter Bala, Feb 16 2024
Equals Sum_{k >= 0} 1/((k + 1)*(2*k + 1)*binomial(2*k, k)). See Catalan, Section 21, equation 30. - Peter Bala, Aug 14 2024
EXAMPLE
1.096622711232150957648276777764...
MATHEMATICA
RealDigits[Pi^2/9, 10, 110][[1]] (* G. C. Greubel, Feb 17 2017 *)
PROG
(PARI) default(realprecision, 110); Pi^2/9 \\ G. C. Greubel, Feb 17 2017
(Sage) numerical_approx(pi^2/9, digits=120) # G. C. Greubel, Jun 02 2021
CROSSREFS
KEYWORD
AUTHOR
Eric W. Weisstein, Oct 31 2004
STATUS
approved