There is a family of generalized continued fractions. I'd like to use the notation $$c = a_0 + {b_0\over a_1 + {b_1\over a_2 + ... }} $$ in accordance with the Angell-article (see below).
The following family of generalized continued fraction gives rational or irrational numbers depending on parametrization: $$\begin{array} {r|r} \begin{matrix}&a_k=\\b_k=&\end{matrix} &&0+k & 1+k & 2+k & 3+k & \cdots \\ \hline 0+k && 0 & 1 & 2 &3 & \cdots \\ \hline 1+k && {1\over e-1} & {1\over e-2} & {1\over 2e-5} & {1\over 6e-16} & \cdots \\ \hline 2+k && {1\over 1} & -{1e-1\over 0e-1} & -{1e-2\over 1e-3}& -{2e-5\over 4e-11} & \cdots \\ \hline 3+k && {4\over 3} & {2 \over 1} & -{0e-2\over 1e-2}& -{2e-6\over 3e-8} & \cdots \\ \hline 4+k && {21\over 13} & {9 \over 4} & {3 \over 1} & -{3e-6\over 2e-6} & \cdots \\ \hline \cdots & & \cdots \end{array}$$
I've found that heuristically, using wolframalpha for support in the evaluation.
David Angell describes that family and gives that heuristic the analytic background (at least for the rational results if I got this correctly) see: David Angell - A family of continued fractions (2010) Journal of Number Theory 130 , pg. 904-911 (Elsevier), online: "paywall"
A somewhat larger table is at my mathpages - GenContFrac