Thank you, Benjamin, for the explanation. This helps a lot.
I still need to look at some more examples to get a feel for the quasipolynomials and their counting. Im guessing that the periodicity somehow conserves the fact that we have 'large' areas without adding new lattice points to \lambda*P ...

I want to investigate rational dilates of rational polytopes. I am aware that the Ehrhart-Polynomial of such polytopes is a quasipolynomial, i.e. the coefficients are periodic functions c_i(lambda), depending on the (rational) scaling factor lambda. Below the output for the function EHRHART_QUASI ...