On methods for dealing with recursively defined sequences

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.

I doubt that there is a general theory, but for this particular example there is a different way. Let $b_n = 4^{-(n+1)} a_n$, then it is easy to check that $b_1 = 1/2$ and $b_{n+1} = \frac{1-\sqrt{1-b_n}}{2}$, i.e., that it really boils down to the asymptotic behavior of the iterates of $1/2$ under the function $f(b) = \frac{1-\sqrt{1-b}}{2}$.
Now it is another fortunate coincidence that $f$ happens to be conjugate to the inverse of the Chebyshev polynomial $T(z) = 2 z^2 - 1$ via the change of coordinates $z = 1-2b$. This leads to the identity $b_n = \frac{1-\cos (2^{-n} \pi)}{2}$ and the Taylor expansion $\cos z = 1 - \frac{z^2}{2} + O(z^4)$ gives the desired result.