THE NUMBER OF GRIDPOINTS ON HYPERPLANE SECTIONS OF THE d-DIMENSIONAL CUBE

We deduce a formula for the exact number of gridpoints (i.e., elements of Z(d)) in the extended d-dimensional cube nC(d) = [-n, +n](d) on intersecting hyperplanes. In the special case of the hyperplanes {x is an element of R-d vertical bar x(1) + ... + x(d) = b}, b is an element of Z, these numbers can be written as a finite sum involving products of certain binomial coefficients. Furthermore, we consider the limit as n tends to infinity which can be expressed in terms of Euler-Frobenius numbers. Finally, we state a conjecture on the asymptotic behaviour of this limit as the dimension d tends to infinity.