Let $L_{m,n} \subset \mathbb{R}^2$ be a rectangle given by $[m,0]\times[0,n]$ with $m,n$ positive integers. Define $N(m,n)$ to be the number of subdivisions of $L_{m,n}$ into lattice triangles of area $1/2$. Here a lattice triangle is a triangle with vertices on the lattice $\mathbb{Z}^2\subset \mathbb{R}^2$. Is there an explicit formulae for $N(m,n)$?
For example, $N(1,1)=2$, $N(1,2)=6$...
Thank you very much in advance.