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What is $\sum\limits_{n=0}^\infty \frac1{(3n+1)^2}$?

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    The sum is easily expressed in terms of the Lerch transcendent: $\dfrac19 \Phi\left(1,2,\dfrac13\right)$. There is an alternative expression in terms of the trigamma function: $\dfrac19\psi^{(1)}\left(\dfrac13\right)$, but deriving that form is a bit more complicated to do...2012-02-08
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    The title and the question are asking two different things: the question asks for squares of (numbers that are of the form $3n+1$), whereas I at least parsed the title as asking for (squares of numbers) that are of the form $3n+1$; the sum in (my interpretation of) the title has an explicit elementary form, as opposed to the sum of the question...2012-02-08
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    See http://mathoverflow.net/questions/87348/sum-of-reciprocals-of-squares-of-integers-congruent-to-1-mod-3 by, one assumes, the same person.2012-02-08

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