More generally, if $1\le k\le N-1$ is an integer, where $N$ is a positive interger, $$S_{N,k} := \sum_{n=0}^\infty\biggl( \frac{1}{(N n+N-k)^2} + \frac{1}{(N n+k)^2} \biggr) = \frac{\pi^2}{N^2\sin^2(\pi k/N)}.$$ Your sum is $S_{10,1}-S_{10,3}$.
Brendan McKay
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