I am pretty sure I have seen this somewhere before. The problem states
$$\sum_{n=0}^{\infty}\frac{(-1)^n}{2n+1}\sum_{k=0}^{2n}\frac{1}{2n+4k+3}=\frac{3\pi}{8}\log \left(\frac{1+\sqrt5}{2} \right)-\frac{\pi}{16}\log 5$$
I have no idea how to tackle it. I, basically, do not know how to tackle problems involving double summation. Any help would be useful.
Addendum: I bumped into it today in my archives and it dates back in 2017.