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

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, 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, 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

Offset: 1

Author

R. J. Mathar, Jul 20 2012

Keywords

• cons
• nonn

Comments

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

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

Examples

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

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Hurwitz Zeta Function.
• Eric Weisstein's World of Mathematics, Polygamma Function.

Crossrefs

Cf. A016777, A086724, A214549, A294967.

Programs

Maple

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

Mathematica

RealDigits[ PolyGamma[1, 1/3]/9, 10, 105] // First (* Jean-François Alcover, Feb 11 2013 *)

PARI

zetahurwitz(2,1/3)/9 \\ Charles R Greathouse IV, Jan 30 2018

PARI

sumpos(n=0,1/(3*n+1)^2) \\ Charles R Greathouse IV, Jan 30 2018

Formula

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

From Amiram Eldar, Oct 02 2020: (Start)

Equals Integral_{0..1} log(x)/(x^3-1) dx = Integral_{1..oo} x*log(x)/(x^3-1) dx.

Equals 4*Pi^2/27 - A294967. (End)

Extensions

More terms from Jean-François Alcover, Feb 11 2013