I stumbled across a problem that involves the following recurrence relation (also with different values for the $\frac{2}{3}$):
$$ p(x+1) = p(x) + \frac{2} {3p(x)}\\ \text {or in alternate form}\\ p(x+1) =\frac{p(x)^2 +\frac2 3}{p(x)} $$
I have no problem finding closed form for the normal linear recurrence relations, but for one that includes fractions like this one does I don't even have a clue where to start.