The periodic-orbit theory of the density-of-states and the closed-orbit theory of atomic absorption spectra relate properties of a quantum system to properties of periodic or of closed orbits of a system. In these theories, every return of an orbit to the initial point makes a contribution, so in general for each orbit an infinite number of terms must be computed and summed. We show that the term asising from the nth return of an orbit can be calculated from properties of the orbit on its first return.