Unverified Families

The following is a list of functions which appear to be families of continued fractions, because a computer program written by the author in the summer of 1999 showed that for small values for the variables in M, the period of $\sqrt{M}$ grew linearly. These results would be highly unlikely if M where in fact not a family of continued fractions.

Let F = 4a + 1, let j = 1 when a = 1, and let j = 2 when a > 1.
$$\begin{align*}M & = (4aF^k + 1)^2 + 4F^k, \qquad \ell = 10k + 5\\ M & = (2aF^k... ... 8k + 4j - 4\\ M & = (aF^k -1)^2 - F^k, \qquad \ell = 4k + 4j - 6 \end{align*}$$
Where E = 4a - 1,
$$\begin{align*}M & = (4aE^k - 1)^2 + 4E^k, \qquad \ell = 11k + 5\\ M & = (2aE^k... ...ad \ell = 6k + 4\\ M & = (aE^k + 1)^2 - E^k, \qquad \ell = 3k + 2 \end{align*}$$
Also, where p = 0 for a= 1 and p = 1 for a> 1,
$$\begin{align*}M & = (4aE^{2k} + E - 1)^2 + 4E, \qquad \ell = 16k - 4\\ M & = (... ... 4\\ M & = (2aE^{k+1} - E + 1)^2 + 4E, \qquad \ell = 16k + 8p + 4 \end{align*}$$
Now let, $G = 2^{d} \cdot a - 1$, with d > 2; then,
$$\begin{align*}M & = (4G^{2k} - G - 1)^2 - 4G, \qquad \ell = 32k\\ M & = (4G^{2... ...= 16k\\ M & = (2G^{2k + 1} - G - 1)^2 - 4G, \qquad \ell = 16k + 4 \end{align*}$$
Letting $H = 2^{d} \cdot a + 3$ where d > 2, one finds,
$$\begin{align*}M & = (4H^{2k} - H - 1)^2 - 4H, \qquad \ell = 24k - 2\\ M & = (4... ...= 8k - 2\\ M & = (2H^{k+1} + H - 1)^2 + 4H, \qquad \ell = 16k + 8 \end{align*}$$
Let $J = 2^{d} \cdot a - 3$ with d > 2; then,
$$\begin{align*}M & = (4J^{2k} - J + 1)^2 + 4J, \qquad \ell = 56k - 6\\ M & = (4... ...k - 4\\ M & = (2J^{2k + 1} + J - 1)^2 + 4J, \qquad \ell = 16k + 2 \end{align*}$$
Where $N = 2^{d} \cdot a + 1$ with d > 2, one has
$$\begin{align*}M & = (4aN^{k + 1} - N - 1)^2 - 4N, \qquad \ell = 16(k + 1)\\ M ... ...\\ M & = (2aN^{k + 1} - N - 1)^2 - 4N, \qquad \ell = 16k + 4j + 8 \end{align*}$$
Looking at G again,
$$\begin{align*}M & = (2^{d} \cdot aG^{k} - 2^{d-2})^2 + 2^{d} \cdot G^{k}, \qqua... ...3k + 1\\ M & = (aG^{k} + 2^{d-2})^2 - L^{k}, \qquad \ell = 3k + 2 \end{align*}$$
Similarly with N,
$$\begin{align*}M & = (2^{d} \cdot aN^k + 2^{d-2})^2 + 2^d \cdot N^k, \qquad \ell... ...k + 1\\ M & = (aN^k - 2^{d-2})^2 - N^k, \qquad \ell = 4k + 4j - 6 \end{align*}$$
Some more families involving G:
$$\begin{align*}M & = (2^d \cdot aG^{2k} + G - 1)^2 + 4G, \qquad \ell = 16k - 4\\... ...d - 2\\ M & = (2aG^k - G + 1)^2 + 4G, \qquad \ell = 16k + 4j - 12 \end{align*}$$
The sort of patterns above can also be seen in N:
$$\begin{align*}M & = (2^d \cdot aN^k - N - 1)^2 - 4N, \qquad \ell = 8k\\ M & = ... ...eq d-2\\ M & = (2aN^k + N + 1)^2 - 4N, \qquad \ell = 16k + 4j - 8 \end{align*}$$
Though the following continued fractions have only one variable, their growth rate is significant (for two of them), so it should be useful to mention them.
$$\begin{align*}M & = (2 \cdot 3^{3k} - 26)^2 + 108, \qquad \ell = 16k - 8\\ M &... ... M & = (2 \cdot 3^{3k + 2} - 26)^2 + 108, \qquad \ell = 48k + 20 \end{align*}$$