I was "playing with $\pi$" trying to look at it in different numeral systems and it's not so hard to obtain $\pi$ base $2$ or $3$ or even $\varphi=\frac{\sqrt{5}+1}{2}$, using Maclaurin series of $\tan^{-1}$ at $\frac{1}{\sqrt{2}}$ and $\frac{1}{\sqrt{3}}$ and $\cos^{-1}$ at $\frac{\sqrt{5}+1}{4}$ respectively, to an arbitrary precision.
In base 3 there are at least two systems of digits: $\{0,1,2\}$ and $\{0,1,-1\}$ and the latter looks "more symmetric" to me.
So I wonder
if there's $a_n$ such that $\pi=\sum\limits_{n=-1}^{\infty} a_ne^{-n}$ where $a_n\in\{0,1,-1\}\ \forall n$
or how to obtain these $a_n$ by computing.
The few values of $\pi_e$ (i.e. $a_n$) might be
1,0;1,0,1,0,1,-1,1,1,1,0,1,0,-1,0,0,-1,1,0,-1,1,0,-1,0,-1,0,-1,0,-1,0,0,1,...
(yes, this representation is ambiguous).
But I'm stuck to find a $\pi_e$ as a series, maybe I should rearrange $e^x$ Maclaurin series somehow?
P.S. I've found $\pi$ base $e$ using $\{0,1,2\}$ digits https://oeis.org/A050948 but it doesn't help much.