A100044 Decimal expansion of Pi^2/9.

1, 0, 9, 6, 6, 2, 2, 7, 1, 1, 2, 3, 2, 1, 5, 0, 9, 5, 7, 6, 4, 8, 2, 7, 6, 7, 7, 7, 7, 6, 4, 0, 1, 6, 7, 9, 2, 8, 1, 2, 6, 3, 3, 2, 6, 7, 4, 7, 1, 1, 9, 8, 9, 5, 8, 4, 9, 0, 3, 7, 2, 1, 5, 2, 9, 1, 3, 3, 3, 8, 3, 1, 3, 6, 0, 2, 1, 3, 3, 9, 1, 5, 8, 8, 9, 0, 8, 5, 9, 3, 3, 7, 4, 6, 5, 0, 5, 8, 0, 3, 5, 3

Offset: 1

Author

Eric W. Weisstein, Oct 31 2004

Keywords

• nonn
• cons
• easy

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

Examples

			1.096622711232150957648276777764...
		

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.
• Index entries for transcendental numbers.

Crossrefs

Cf. A000120, A007310, A008833, A019670, A086463, A091999, A104777, A120325, A112093, A112094, A130549, A130550.

Programs

Mathematica

RealDigits[Pi^2/9, 10, 110][[1]] (* G. C. Greubel, Feb 17 2017 *)

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

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