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Decimal expansion of Pi^2/9.
17

%I #89 Aug 22 2024 04:47:22

%S 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,

%T 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,

%U 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

%N Decimal expansion of Pi^2/9.

%C The Dirichlet L-series for the principal character mod 6 (which is A120325 shifted left) evaluated at 2. - _R. J. Mathar_, Jul 20 2012

%C Equals the asymptotic mean of the abundancy index of the numbers coprime to 6 (A007310). - _Amiram Eldar_, May 12 2023

%D 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.

%D L. B. W. Jolley, Summation of Series, Dover, 1961.

%H G. C. Greubel, <a href="/A100044/b100044.txt">Table of n, a(n) for n = 1..5000</a>

%H Jean-Paul Allouche and Jeffrey Shallit, <a href="https://doi.org/10.1007/BFb0097122">Sums of digits and the Hurwitz zeta function</a>, in: K. Nagasaka and E. Fouvry (eds.), Analytic Number Theory, Lecture Notes in Mathematics, Vol. 1434, Springer, Berlin, Heidelberg, 1990, pp. 19-30.

%H Eugène-Charles Catalan, <a href="https://www.persee.fr/doc/marb_0776-3123_1867_num_33_1_1705">Mémoire sur la transformation des séries et sur quelques intégrales définies</a>, Mémoires de l'Académie royale de Belgique, 1867, Vol. 33, pp. 1-50.

%H R. J. Mathar, <a href="https://arxiv.org/abs/1008.2547">Table of Dirichlet L-series and prime zeta modulo functions for small moduli</a>, arXiv:1008.2547 [math.NT], 2010-2015, Table 22.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DigitSum.html">Digit Sum</a>.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.

%F 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]

%F 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

%F 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

%F Equals A019670^2. - _Michel Marcus_, May 19 2014

%F Equals 2*A086463 = 2*Sum_{n>=1} 1/A091999(n)^2, equivalent to the formula of 2012 above. - _Alexander R. Povolotsky_, May 20 2014

%F 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

%F 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

%F Equals Integral_{x=0..oo} log(x)/(x^6 - 1) dx. - _Amiram Eldar_, Aug 12 2020

%F Equals Sum_{k>=1} A000120(k) * (2*k+1)/(k^2*(k+1)^2) (Allouche and Shallit, 1990). - _Amiram Eldar_, Jun 01 2021

%F Equals Integral_{x=0..1} log(1+x+x^2)/x dx (Aubonnet). - _Bernard Schott_, Feb 04 2022

%F Equals Sum_{k>=1} A008833(k)/k^4. - _Amiram Eldar_, Jan 25 2024

%F 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

%F 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

%e 1.096622711232150957648276777764...

%t RealDigits[Pi^2/9, 10, 110][[1]] (* _G. C. Greubel_, Feb 17 2017 *)

%o (PARI) default(realprecision, 110); Pi^2/9 \\ _G. C. Greubel_, Feb 17 2017

%o (Sage) numerical_approx(pi^2/9, digits=120) # _G. C. Greubel_, Jun 02 2021

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

%K nonn,cons,easy

%O 1,3

%A _Eric W. Weisstein_, Oct 31 2004