In this study, we propose a non-negative integer-valued model based on the sum of Poisson-Lindley and geometric distributions. We show that it corresponds to the weighted geometric distribution and also a special mixture of two negative binomial distributions with certain parameters. The main statistical properties of the new distribution are studied comprehensively, including estimation of the model parameter. A new count regression analysis is introduced by using the new distribution. Finally, we provide some applications on practical data sets.