Examples
If P is a d-dimensional unit hypercube whose vertices are the integer lattice points all of whose coordinates are 0 or 1, then the t-fold dilation of P is a cube with side length t, containing (t + 1)d integer points. That is, the Ehrhart polynomial of the hypercube is L(P,t) = (t + 1)d.
Many other figurate numbers can be expressed as Ehrhart polynomials. For instance, the square pyramidal numbers are given by the Ehrhart polynomials of a square pyramid with an integer unit square as its base and with height one; the Ehrhart polynomial in this case is (t + 1)(t + 2)(2t + 3)/6.
Read more about this topic: Ehrhart Polynomial
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