Define $a_1=8$ and $a_n=\frac{4^{n+1}-2^{n+2}\sqrt{4^n-a_{n-1}}}{2}$ for $n\geq 2$.

By means of harmonic analysis methods it can be shown that $a_n$ converges to $\pi^2$ (this being the first eigenvalue of for the string of unit length and Dirichlet bc).

Does anyone know how to prove this from a different approach?

More generally, I would like to hear if there is some sort of theory (or theories) dealing with sequences as the one above; to be a bit more precise: what can be said of a sequence defined by $a_n=f_m(a_{n-1})$ where $f_n$ are functions depending on $n$ in a (fairly simple and regular) way.