That's kind of cute: Let $ S(x) = \sum_{n \geq 0} S_n x^n$ be the generating function for $ S_n$. Then $ S(x) = x + \sum_{n \geq 2} (S_{n - 1} + S_{n - 2} + F_n)x^n = x + xS(x) + x^2S(x) + F(x)$, where $ F(x) = \frac {x}{1 - x - x^2}$ is the generating function for the Fibonacci numbers. This immediately gives $ S(x) = \frac {x}{(1 - x - x^2)^2}$, so $ xS(x) = F(x)^2$ and thus $ S_n = \sum_{k = 0}^{n + 1} F_k F_{n + 1 - k}$.

Also \[ S_n = \frac{nF_{n + 1} + 2(n + 1)F_n}{5}.\]