BALKAN 1995 Problem 4

Let $n$ be a positive integer and $\mathcal S$ be the set of points $(x, y)$ with $x, y \in \{1, 2, \ldots , n\}$ . Let $\mathcal T$ be the set of all squares with vertices in the set $\mathcal S$ . We denote by $a_k$ ( $k \geq 0$ ) the number of (unordered) pairs of points for which there are exactly $k$ squares in $\mathcal T$ having these two points as vertices. Prove that $a_0 = a_2 + 2a_3$ .

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