The key is to consider how many squares have their vertices on the perimeter of a given $(n+1) \times (n+1)$ array whose sides are parallel to the sides of the array.

The diagram shows that there are $n$ such squares. There are $(k+1-n)^2$ such arrays. So the total number of squares is $k \cdot 1^2 + (k-1)^2 \cdot 2 + \cdots + 1 \cdot k^2 = \sum_{1}^{k}(k+1-i)i^2 = \frac{(k+1)k(k+1)(2k+1)}{6} - \frac{k^2(k+1)^2}{4} = \frac{k(k+1)2(k+2)}{12}$.