II. (50 points) Among the subsets of the set $S_{n}=\{1,2, \cdots, n\}$, those that do not contain two consecutive natural numbers are called "good subsets". How many good subsets are there in $S_{n}$?
Let $S_{n}$ have $a_{n}$ good subsets.
When $n=1$, $S_{n}$ has subsets $\varnothing,\{1\}$ which are good subsets, $a_{1}=2$. When $n=2$, $a_{2}=3$.
For $n \geqslant 3$, let $M$ be a good subset of $S_{n}$.
If $n \notin M$, then $M$ is also a good subset of $S_{n-1}$, and conversely, a good subset of $S_{n-1}$ is also a good subset of $S_{n}$.
If $n \in M$, then by the definition of a good subset, $n-1 \notin M$, so the subset $M^{\prime}\left(M=M^{\prime} \cup\{n\}\right)$ obtained by removing $n$ from $M$ must be a good subset of $S_{n-2}$, and conversely, a good subset of $S_{n-2}$ is also a good subset of $S_{n}$. Thus, $\left\{\begin{array}{l}a_{1} \neq 2, a_{2}=3 \\ a_{n}=a_{n-1}+a_{n-2}\end{array}\right.$.
Solving the characteristic equation yields $x_{1}=\frac{1+\sqrt{5}}{2}, x_{2}=\frac{1-\sqrt{5}}{2}$. Therefore, $a_{n}=\frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n+2}-\left(\frac{1-\sqrt{5}}{2}\right)^{n+2}\right]$.