Levesque and Rhin's Families

In their paper [3], Levesque and Rhin obtain some families of continued fractions in the same way as Berstein, and use the same inductive method of proof to show that the functions they have are in fact families of continued fractions. Their families are listed below, and again, the list appears in the same format as it did in [3].

Letting E = 4a - 1, they obtained the families:
$$\begin{align*}M & = (4aE^k - 2a)^2 + 8aE^k, \qquad \ell = 16k\\ M & = (4aE^k +... ... M & = (aE^k - a)^2 + aE^k, \qquad a \neq 1, \qquad \ell = 6k + 2 \end{align*}$$
Let Q = 4a² - 1. They obtained the following two families associated with Q:
$$\begin{align*}M & = (aQ^k + a)^2 - Q^k, \qquad \ell = 3k + 2\\ M & = (aQ^k - a)^2 + Q^k, \qquad \ell = 3k + 1 \end{align*}$$
Finally, they obtained the families:
$$\begin{align*}M & = (2^{k+3} - 3)^2 - 8, \qquad \ell = 4k + 6\\ M & = (2^{k + ... ...qquad \ell = 5k\\ M & = (2^{k+1} - 1)^2 + 8, \qquad \ell = 5k - 4 \end{align*}$$