Exact ranges of certain generalized polynomials with rational parameters

For fα(n)=⌊α2n⌋−⌊α⌊αn⌋⌋, n∈ℕ, f_\alpha(n) = \lfloor \alpha^2 n \rfloor - \lfloor \alpha \lfloor \alpha n \rfloor \rfloor, \quad n \in \mathbb{N}, we give an exact finite description of Range(f_α) for every positive rational parameter α. In particular, this settles all supra-unitary rational cases and proves Conjecture 3.17 from the authors’ previous work [1].