Reverse continued fractions Basic continued fractions arise from recurrence relations such as:
$$
n = a + \frac{b}{n},
$$
This gives rise to the continued fraction:
$$
n = a + \frac{b}{a+\frac{b}{a+\frac{b}{a+...}}}.
$$
What about relations such as:
$$
n = a + \frac{n}{b}?
$$
Do these give rise to things like:
$$
n = a + \frac{a+\frac{a+\frac{a+\frac{a+\frac{a+...}{b}}{b}}{b}}{b}}{b}?
$$
Of course, one could just solve the original equation as:
\begin{align}
n &= a + \frac{n}{b} \\
n(1-b^{-1}) &= a \\
n &= \frac{a}{1-b^{-1}}.
\end{align}
But couldn't the above "reverse continued fraction" be generalized like the usual one to create interesting structures and definitions of known constants? For instance, one can obtain this strange result for $n=a + \frac{n}{b²}$ by setting $a=1$ and $b=\sqrt{2}^{-1}$:
$$
-1 = 1 + \frac{1+\frac{1+\frac{1+\frac{1+\frac{1+...}{\sqrt{2}^{-1}}}{\sqrt{2}^{-1}}}{\sqrt{2}^{-1}}}{\sqrt{2}^{-1}}}{\sqrt{2}^{-1}}?
$$
Is this known/of any particular interest?

Note that $$n = a + \frac{a+\frac{a+\frac{a+\frac{a+\frac{a+...}{b}}{b}}{b}}{b}}{b}=a+\frac ab + \frac a{b^2}+ \frac a{b^3} +\ldots$$
gives you a geometric series.  If the constants vary it is still just another way of writing an infinite sum, so doesn't give anything new.
