There is apparently a curious connection between Euler-Mascheroni constant $\gamma$ and $e$ in the form of an infinite series and continued fraction:
$$e \gamma=e \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n!~n}-\cfrac{1}{2-\cfrac{1}{4-\cfrac{4}{6-\cfrac{9}{8-\cfrac{16}{10-\cdots}}}}}$$
As can be seen, the partial denominators and numerators have the form $2n$ and $-n^2$ respectively.
How can we prove this? Might it be a useful method to compute $\gamma$?