21.3.5 ** In a triangle with integer side lengths, given one side length as $n$, and the other two side lengths do not exceed $n$, congruent triangles are counted as one. Find the number of such triangles.
Let the side length $c \leqslant b \leqslant n$. We discuss the following two cases:
(1) When $n=2 k$, for $c=1,2, \cdots, k$, $b$ has $1,2, \cdots, k$ values respectively. For $c>k$, $c$ can take any two values (which can be the same) from $k+1, \cdots, n$, so there are a total of $1+2+\cdots+k+k+\mathrm{C}_{k}^{2}=k^{2}+$ $k=\frac{n^{2}+2 n}{4}$ triangles.
(2) When $n=2 k-1$, similarly, the number of triangles is
$$
1+2+\cdots+(k-1)+\mathrm{C}_{k}^{2}+k=k^{2}=\left(\frac{n+1}{2}\right)^{2} .
$$

In summary, the number of triangles is $\left[\frac{(n+1)^{2}}{4}\right]$.