Example 4 Given the set $\{1,2,3,4,5,6,7,8,9,10\}$. Find the number of subsets of this set that have the following property: each subset contains at least 2 elements, and the absolute difference between any two elements in each subset is greater than 1.
(1996, Shanghai High School Mathematics Competition)
Let $a_{n}$ be the number of subsets of the set $\{1,2, \cdots, n\}$ that have the given property. Then, the subsets of $\{1,2, \cdots, k, k+1, k+2\}$ that have the given property can be divided into two categories: the first category does not contain $k+2$, and there are $a_{k+1}$ such subsets; the second category contains $k+2$, and these subsets are either the union of a corresponding subset of $\{1,2, \cdots, k\}$ with $\{k+2\}$, or the union of a single-element subset of $\{1,2, \cdots, k\}$ with $\{k+2\}$, totaling
\[
\begin{array}{l}
a_{k}+k \text { subsets. } \\
\text { Therefore, } a_{k+2}=a_{k+1}+a_{k}+k . \\
\text { Clearly, } a_{3}=1, a_{4}=3 . \\
\text { Thus, } a_{5}=7, a_{6}=14, a_{7}=26, \\
a_{8}=46, a_{9}=79, a_{10}=133 .
\end{array}
\]