I recently asked the question at bottom. This is an update.
From googling I learned that an n x n matrix whose rows are successive cyclic rotated versions of each other are called "circulant" matrices (a term I'd heard but never bothered to learn about before) — and that these matrices are actually very interesting!!! (See for instance this article: <https://www.ams.org/notices/201203/rtx120300368p.pdf>.
It is apparently an open problem to determine which integers can be the determinants of integer circulant matrices — see for instance this article: <https://projecteuclid.org/download/pdf_1/euclid.ijm/1256047802>. Perhaps the additional condition I somewhat arbitrarily imposed, that the row sum a(0) + ... + a(n-1) = 1, might allow solution of the problem in this special case; maybe not. A subproblem might be to reduce entries of a circulant matrix modulo a prime p and then ask what the determinant can be modulo p. Maybe that question is more tractable. —Dan ----- 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 ??? -----