It is well known that plurigenus does not vanish for a sufficiently large multiple on a canonical threefold over $mathbb{C}$. There is Reid Fletcher formula for plurigenus. But, unlike in the case of surface of general type, it is not easy to compute plurigenus. In this paper, we in-duce a different version of Reid-Fletcher formula and show that the constant term in the induced formula has periodic properties. Using these properties we have an application to nonvanishing of plurigenus.