Let Z1, Z2, . . . be a sequence of independent Bernoulli trials with constant success and failure probabilities p = Pr(Zt = 1) and q = Pr(Zt = 0) = 1 − p, respectively, t = 1, 2, . . . . For any given integer k ≥ 2 we consider the patterns \({\mathcal{E}_{1}}\): two successes are separated by at most k−2 failures, \({\mathcal{E}_{2}}\): two successes are separated by exactly k −2 failures, and \({\mathcal{E}_{3}}\) : two successes are separated by at least k − 2 failures. Denote by \({ N_{n,k}^{(i)}}\) (respectively \({M_{n,k}^{(i)}}\)) the number of occurrences of the pattern \({\mathcal{E}_{i}}\) , i = 1, 2, 3, in Z1, Z2, . . . , Zn when the non-overlapping (respectively overlapping) counting scheme for runs and patterns is employed. Also, let \({T_{r,k}^{(i)}}\) (resp. \({W_{r,k}^{(i)})}\) be the waiting time for the r − th occurrence of the pattern \({\mathcal{E}_{i}}\), i = 1, 2, 3, in Z1, Z2, . . . according to the non-overlapping (resp. overlapping) counting scheme. In this article we conduct a systematic study of \({N_{n,k}^{(i)}}\), \({M_{n,k}^{(i)}}\), \({T_{r,k}^{(i)}}\) and \({W_{r,k}^{(i)}}\) (i = 1, 2, 3) obtaining exact formulae, explicit or recursive, for their probability generating functions, probability mass functions and moments. An application is given.