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A214550 Decimal expansion of Sum_{n>=0} 1/(3*n+1)^2.

%I A214550 #34 Feb 16 2025 08:33:18
%S A214550 1,1,2,1,7,3,3,0,1,3,9,3,6,3,4,3,7,8,6,8,6,5,7,7,8,2,3,3,3,2,1,3,9,0,
%T A214550 7,0,6,7,2,4,3,2,2,6,7,9,9,2,0,1,0,8,6,8,2,4,3,7,9,6,4,8,0,0,0,9,2,3,
%U A214550 3,5,7,0,1,3,9,3,8,9,8,3,8,6,3,0,5,8,2,5,4,0,7,9,1,3,7,7,5,4,6,6,2,0,1,1,8
%N A214550 Decimal expansion of Sum_{n>=0} 1/(3*n+1)^2.
%C A214550 Sum over the inverse squares of A016777. Dirichlet series Sum_{n>=1} A079978(n-1)/n^s at s=2.
%C A214550 This is also (1/9)*Zeta(2, 1/3) = (1/9)*Psi(1, 1/3) with the Hurwitz zeta function Zeta(s, a) and the Polygamma function Psi(n, z). See the programs. - _Wolfdieter Lang_, Nov 12 2017
%H A214550 G. C. Greubel, <a href="/A214550/b214550.txt">Table of n, a(n) for n = 1..10000</a>
%H A214550 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HurwitzZetaFunction.html">Hurwitz Zeta Function</a>.
%H A214550 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PolygammaFunction.html">Polygamma Function</a>.
%F A214550 Equals (A086724 + A214549)/2 because the sequence represented by A079978 (with offset 1) is the average of A011655 and A102283.
%F A214550 From _Amiram Eldar_, Oct 02 2020: (Start)
%F A214550 Equals Integral_{0..1} log(x)/(x^3-1) dx = Integral_{1..oo} x*log(x)/(x^3-1) dx.
%F A214550 Equals 4*Pi^2/27 - A294967. (End)
%e A214550 1.1217330139363437868657... = 1/1^2 + 1/4^2 + 1/7^2 + 1/10^2 + 1/13^2 + ...
%p A214550 evalf(Psi(1,1/3)/9);
%t A214550 RealDigits[ PolyGamma[1, 1/3]/9, 10, 105] // First (* _Jean-François Alcover_, Feb 11 2013 *)
%o A214550 (PARI) zetahurwitz(2,1/3)/9 \\ _Charles R Greathouse IV_, Jan 30 2018
%o A214550 (PARI) sumpos(n=0,1/(3*n+1)^2) \\ _Charles R Greathouse IV_, Jan 30 2018
%Y A214550 Cf. A016777, A086724, A214549, A294967.
%K A214550 cons,nonn
%O A214550 1,3
%A A214550 _R. J. Mathar_, Jul 20 2012
%E A214550 More terms from _Jean-François Alcover_, Feb 11 2013