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Decimal expansion of Pi^2/15.
15

%I #40 Feb 04 2024 03:24:59

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

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

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

%N Decimal expansion of Pi^2/15.

%H László Tóth, <a href="http://math.colgate.edu/~integers/w98/w98.pdf">Linear combinations of Dirichlet series associated with the Thue-Morse sequence</a>, Integers: Electronic Journal of Combinatorial Number Theory, Vol. 22 (2022), #A98; <a href="https://arxiv.org/abs/2211.13570">arXiv preprint</a>, arXiv:2211.13570 [math.NT], 2022.

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

%F See Mathematica code.

%F Equals Gamma(4)*zeta(4)/Pi^2 = zeta(4)/zeta(2) = A013662/A013661 = Product_{p prime} (p^2/(p^2+1)). - _Stanislav Sykora_, Oct 21 2014

%F Equals (1/10) * Sum_{n >= 0} (-1)^n*( 1/(n + 1/3)^2 - 1/(n + 2/3)^2 ). - _Peter Bala_, Oct 31 2019

%F Equals Sum_{k>=1} A008836(k)/k^2. - _Amiram Eldar_, Jun 23 2020

%F Equals (1/10) * Sum_{k>=1} (5*t(k-1) + 3*t(k))/k^2, where t(k) = A010060(k) (Tóth, 2022). - _Amiram Eldar_, Feb 04 2024

%e 0.65797362673929...

%t RealDigits[N[Sum[1/(n + 0)^2 - 1/(n + 1)^2 + 1/(n + 2)^2 - 1/(n + 3)^2 - 4/(n + 4)^2 - 1/(n + 5)^2 + 1/(n + 6)^2 - 1/(n + 7)^2 + 1/(n + 8)^2 + 4/(n + 9)^2, {n, 1, Infinity, 10}], 90]][[1]]

%t RealDigits[N[Sum[LiouvilleLambda[n]/n^2, {n, 1, Infinity}], 90]][[1]]

%t RealDigits[Pi^2/15,10,120][[1]] (* _Harvey P. Dale_, May 28 2017 *)

%o (PARI) Pi^2/15 \\ _Michel Marcus_, Oct 21 2014

%Y Cf. A000796, A008836, A010060, A013661, A013662, A086463, A191898, A249103.

%Y Cf. A347328, A347329, A347330, A347331.

%K nonn,cons

%O 0,1

%A _Mats Granvik_, Apr 29 2012

%E Offset corrected and more terms added by _Rick L. Shepherd_, Jan 08 2014