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A214550 Decimal expansion of Sum_{n>=0} 1/(3n+1)^2. 7

%I

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

%C Sum over the inverse squares of A016777. Dirichlet series Sum_{n>=1} A079978(n-1)/n^s at s=2.

%C 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 G. C. Greubel, <a href="/A214550/b214550.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HurwitzZetaFunction.html">Hurwitz Zeta Function </a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PolygammaFunction.html"> Polygamma Function</a>.

%F Equals (A086724 + A214549) /2 because the sequence represented by A079978 (with offset 1) is the average of A011655 and A102283.

%e 1.1217330139363437868657... = 1/1^2 + 1/4^2 + 1/7^2 + 1/10^2 + 1/13^2 + ...

%p evalf(Psi(1,1/3)/9) ;

%t RealDigits[ PolyGamma[1, 1/3]/9, 10, 105] // First (* _Jean-Fran├žois Alcover_, Feb 11 2013 *)

%o (PARI) zetahurwitz(2,1/3)/9 \\ _Charles R Greathouse IV_, Jan 30 2018

%o (PARI) sumpos(n=0,1/(3*n+1)^2) \\ _Charles R Greathouse IV_, Jan 30 2018

%K cons,nonn

%O 1,3

%A _R. J. Mathar_, Jul 20 2012

%E More terms from _Jean-Fran├žois Alcover_, Feb 11 2013

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Last modified September 18 23:06 EDT 2020. Contains 337174 sequences. (Running on oeis4.)