Upon further inquiry, I need to amend my previous post. The "..no known procedure for counting lattice points..." refers to the GENERAL polytope.
There are numerous methods to find the number of lattice points for special simplices. Our simplex is in the form conv(0, l*e, l*e2, l*e3) in R^3 where l is any integer. Its basically derived from Pick's Theorem and Ehrhart's Polynomial. Its a nice little little equation.