We study a novel problem of mining significant recurrent rules from a sequence database. Recurrent rules have the form "whenever a series of precedent events occurs, eventually a series of consequent events occurs". Recurrent rules are intuitive and characterize behaviors in many domains. An example is in the domain of software specifications, in which the rules capture a family of program properties beneficial to program verification and bug detection. Recurrent rules generalize existing work on sequential and episode rules by considering repeated occurrences of premise and consequent events within a sequence and across multiple sequences, and by removing the "window" barrier. Bridging the gap between mined rules and program specifications, we formalize our rules in linear temporal logic. We introduce and apply a novel notion of rule redundancy to ensure efficient mining of a compact representative set of rules. Performance studies on benchmark datasets and a case study on an industrial system have been performed to show the scalability and utility of our approach.