Finding Function Representation of Recursive Sequence

Question

I was trying to find one of the roots of $x^2 + 4x + 3 = 0$ by deriving a continued fraction from the recursive formula $x = -3/x - 4$ (every step of the approximation you increase the recursion by one level and remove the $-3/x$ term), when I developed the sudden urge to prove rigorously that it converged to $-3$. So I thought,

1. derive a natural number sequence $(a_n)$ from the sequence of approximations
2. prove the sequence is the real deal through typical induction trick of showing $n = 0$ is equal to first (or second) approximation from the recursion formula, and then plugging the $n^{th}$ term from the sequence formula and showing it's equal to the $(n+1)^{th}$ term in the sequence. Then take $\lim_{n\rightarrow\infty} a_n$ to prove it converges.

I ended up with a sequence $-13/4, -40/13, -121/40$,... but I'm having a hard time converting this into a formula where the variable $n$ is a natural number (I'm including $0$). I can see the pattern, where each term is $-1/(\text{previous numerator})$ and smaller than $-3$, but, idk, am I approaching this the wrong way?