The modeling of the claims frequency distribution is of perennial importance to actuarial practitioners. An important class is the (a,b,1) class of discrete distributions, which encompass the binomial, negative binomial, Poisson, as well as more exotic distributions such as the logarithmic and Sibuya distributions. It is a recursively defined class, but with an explicit modification at point 0. In this talk we outline how the class can be broadened to accomodate more low claim frequency modifications, leading to what we call the (a,b,r) class. This is opportune, as there are some lines of insurance that have idiosyncratic claim frequency distributions at the lower end (not just at 0). After highlighting various statistical attributes and attendant parameter estimators for some select (a,b,r) models, we apply the model to a large (2011) Brazilian auto claims data set for different values of r and compare the resulting model fits.