I posted the following question on MathStackExchange, but I didn't any answer. So please let me post it on MathOverflow.

Let $L_{m,n}\subset\mathbb R^2$ be a rectangle given by $[0,m]×[0,n]$ 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 formula for $N(m,n)$?

For example, $N(1,1)=2$, $N(1,2)=6$...

Thank you very much in advance.