Simple analytical formulas for the number of energy levels in the vibrational polyads are given. These formulas account for the resonances between the vibrational modes, and for the symmetry of the problem, so that the number of states of a particular symmetry type can be computed. The formulas are used to estimate the differential and integral densities of states from the minimum initial information about the molecule. Examples of the vibrational structure of triatomic molecules A(3), tetrahedral molecules AB(4), and linear molecules AB(2) are considered. The analytical formulas are compared to the ab initio results for H-3(+) [J.R. Henderson et al., J. Chem. Phys. 98, 7191 (1993)].