Today I have an arctan sum but I don't know if has a closed form :
$$\sum\limits_{n=1}^{\infty}\arctan~^{2} \left (\frac{1}{2n}\right)$$
Wolfram alpha give see her. $$\sum\limits_{n=1}^{\infty} \text{arctan}~^{2} \left (\frac{1}{2n}\right) ≈ 0.372905$$ I don't have any ideas to start with this amazing summation I'm happy to see your work!
Thanks!