[math-fun] Integer determinant question

The lattice in R^3 generated by the symmetrical vectors { (2,-1,0), (0,2,-1), (-1,0,2) } (whose sum is (1,1,1)) has a fundamental cell whose volume = 7. I was surprised that a 3-fold symmetrical set of vectors would generate a parallelepiped whose volume had volume = 7. So I'm curious: Suppose n integers a(0), ..., a(n-1) have sum a(0) + ... + a(n-1) = 1. Then what integers can be the determinant |M| of the n x n matrix M = (m(i,j)) if m(i,j) = a(i+j) where i+j is calculated modulo n ??? —Dan