Variations on an error sum function for the convergents of some powers of $e$

Abstract: Several years ago the second author playing with different "recognizers of real constants", e.g., the LLL algorithm, the Plouffe inverter, etc. found empirically the following formula. Let $p_n/q_n$ denote the $n$th convergent of the continued fraction of the constant $e$, then $$ \sum_{n \geq 0} |q_n e - p_n| = \frac{e}{4} \left(- 1 + 10 \sum_{n \geq 0} \frac{(-1)^n}{(n+1)! (2n^2 + 7n + 3)}\right). $$ The purpose of the present paper is to prove this formula and to give similar formulas for some powers of $e$.