14. Let the set $S=\{1,2, \cdots, n\}, A$ be an arithmetic sequence with at least two terms, a positive common difference, and each term in $S$. Additionally, adding any other element of $S$ to $A$ does not form an arithmetic sequence with the same common difference as $A$. Find the number of such $A$ (here, sequences with only two terms are also considered arithmetic sequences).
14. Let the number of sets $A$ that satisfy the conditions in the $n$-element set $S=\{1,2, \cdots, n\}$ be $a_{n}$, then $a_{1}=0 ; a_{2}=1$; 
$a_{3}=2$, these two are $A=\{1,3\},\{1,2,3\}$; 
$a_{4}=4$, these four are $A=\{1,3\},\{1,4\},\{2,4\},\{1,2,3,4\}$. 
The $n$-element set $S=\{1,2, \cdots, n\}$ has more sets $A$ than the $(n-1)$-element set $S=\{1,2, \cdots, n-1\}$ by: one with a common difference of $n-1$, one with a common difference of $n-2$, $\cdots$, one with a common difference of $\frac{n}{2}$ (if $n$ is even) or $\frac{n+1}{2}$ (if $n$ is odd), a total of $\left[\frac{n}{2}\right]$ more, so $a_{n}=$ $a_{n-1}+\left[\frac{n}{2}\right]$. Thus, $a_{n}=a_{n-1}+\left[\frac{n}{2}\right], a_{n-1}=a_{n-2}+\left[\frac{n-1}{2}\right], \cdots, a_{2}=a_{1}+\left[\frac{2}{2}\right]$, adding all these equations gives: $a_{n}=$ $\left[\frac{2}{2}\right]+\left[\frac{3}{2}\right]+\cdots+\left[\frac{n}{2}\right]=\left[\frac{n^{2}}{4}\right]$.