Discrete renewal processes are ubiquitous in stochastic phenomenon. In this talk constructing a discrete process where renewals are more (or less) likely during specified seasons is of specific interest. For example thunderstorms in the Southern United States can take place at any time in the year, but are most likely during the summer. Hurricanes, tornadoes, and snowstorms are other meteorological count processes obeying periodic dynamics. Rare disease occurrences, accidental deaths, and animal sightings are non-meteorological examples of count phenomenon following a periodic structure. In this talk a periodic version of classical discrete-time renewal sequences is developed. Discrete time renewal processes are then used to generate stationary count time series. By superimposing or mixing versions of discrete time renewal processes, one can construct models for sequences of counts. The advantage of this method is we can easily archive many desirable properties i.e. exible autocorrelation structure, periodicity, long memory, etc. The methods are used to develop an autocorrelated periodic count model, fitting a stationary count model to a weekly rainfall data set that has binomial marginal distributions.