Euclid meets Popeye: The Euclidean Algorithm for $2\times 2$ matrices

An analogue of the Euclidean algorithm for square matrices of size 2 with integral non-negative entries and strictly positive determinant $n$ defines a finite set $\mathcal{R}(n)$ of Euclid-reduced matrices corresponding to elements of $\{(a, b, c, d) \in \mathbb{N}^4 | n = ab - cd,\ 0 \le c, d < a, b\}$. With Popeye's help[2] on the use of sails of lattices we show that $\mathcal{R}(n)$ contains $\sum{d|n, d^2 \ge n} (d + 1 - n/d)$ elements.