Three. (This question is worth 16 points) 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.


Let $a_{n}$ be the number of subsets of the set $\{1,2, \cdots, n\}$ that have the given property.

The set $\{1,2, \cdots, n, n+1, n+2\}$ has subsets with the given property divided into two categories: the first category of subsets contains the element $n+2$, and there are $a_{n}+n$ such subsets (i.e., the union of each subset of $\{1, 2, \cdots, n\}$ with $\{n+2\}$, as well as $\{1, n+2\},\{2, n+2\}, \cdots,\{n, n+2\}$); the second category of subsets does not contain $n+2$, and there are $a_{n+1}$ such subsets.
Thus, we have $a_{n+2}=a_{n}+a_{n+1}+n$.
Clearly, $a_{3}=1, \quad a_{4}=3$ (i.e., $\{1,3\},\{2,4\}$, $\{1,4\}$ )
$\therefore a_{5}=7, \quad a_{6}=14, \quad a_{7}=26, \quad a_{8}=46$, $a_{9}=79, \quad a_{10}=133$