Let $n$ be an integer greater than $3$. A square of side length $n$ is divided by lines parallel to each side into $n^2$ squares of length $1$. Find the number of convex trapezoids which have vertices among the vertices of the $n^2$ squares of side length $1$, have side lengths less than or equal $3$ and have area equal to $2$
Note: Parallelograms are trapezoids.
To solve the problem, we need to find the number of convex trapezoids with vertices among the vertices of the \( n^2 \) unit squares, side lengths less than or equal to 3, and area equal to 2. We will consider all possible configurations of trapezoids that meet these criteria.

1. **Identify possible side lengths of parallel sides:**
   - Let \( a \) and \( b \) be the lengths of the parallel sides of the trapezoid, with \( a \geq b \).
   - The possible pairs \((a, b)\) that can form a trapezoid with area 2 are:
     - \((1, 1)\)
     - \((2, 1)\)
     - \((3, 1)\)
     - \((2, 2)\)
     - \((\sqrt{2}, \sqrt{2})\)

2. **Calculate the height \( h \) for each pair \((a, b)\):**
   - For \((1, 1)\):
     \[
     \text{Area} = \frac{1}{2} \times (1 + 1) \times h = 2 \implies h = 2
     \]
   - For \((2, 1)\):
     \[
     \text{Area} = \frac{1}{2} \times (2 + 1) \times h = 2 \implies h = \frac{4}{3}
     \]
   - For \((3, 1)\):
     \[
     \text{Area} = \frac{1}{2} \times (3 + 1) \times h = 2 \implies h = \frac{1}{2}
     \]
   - For \((2, 2)\):
     \[
     \text{Area} = \frac{1}{2} \times (2 + 2) \times h = 2 \implies h = 1
     \]
   - For \((\sqrt{2}, \sqrt{2})\):
     \[
     \text{Area} = \frac{1}{2} \times (\sqrt{2} + \sqrt{2}) \times h = 2 \implies h = \frac{2}{\sqrt{2}} = \sqrt{2}
     \]

3. **Evaluate the feasibility of each pair \((a, b)\):**
   - \((1, 1)\): This is a rectangle with area 2. The number of such rectangles is \( (n-1)^2 \).
   - \((2, 1)\): The height \( h = \frac{4}{3} \) is not an integer, so this configuration is not possible.
   - \((3, 1)\): The height \( h = \frac{1}{2} \) is not an integer, so this configuration is not possible.
   - \((2, 2)\): This is a square with area 2. The number of such squares is \( (n-1)^2 \).
   - \((\sqrt{2}, \sqrt{2})\): The height \( h = \sqrt{2} \) is not an integer, so this configuration is not possible.

4. **Count the number of valid trapezoids:**
   - For \((1, 1)\): The number of such rectangles is \( (n-1)^2 \).
   - For \((2, 2)\): The number of such squares is \( (n-1)^2 \).

5. **Sum the counts:**
   \[
   \text{Total number of trapezoids} = (n-1)^2 + (n-1)^2 = 2(n-1)^2
   \]

The final answer is \(\boxed{2(n-1)^2}\)