Dynamic information networks (DIN), which contain evolving objects in the real world and the links among them, are often modeled as a series of static undirected graph snapshots. A community consists of a group of well-connected objects in an information network. In a DIN, there is often a community whose size increases over time but its members always keep well-connected during that period of time. The evolving trajectory of such a community over time forms a sequence of the community on several snapshots of the DIN, which is termed a lasting enlarging community sequence in this study. It is meaningful to search for lasting enlarging community sequences in a DIN. However, no previous research has paid attention to such community sequences. This study formally defines the q-based lasting enlarging community sequence (qLEC) in a DIN by combining set inclusion with the triangle-connected k-truss model. A two-phase search algorithm is developed, which includes computing candidate vertex sets of communities from the beginning to the end of the time window and performing community sequence search from the end to the beginning of the time window. This study also provides optimization strategies based on early termination and TCP index compression to reduce time and space costs. Sufficient experiments demonstrate that the qLEC model has specific practical significance compared to existing dynamic community models. The two-phase search algorithm effectively finds qLEC-based lasting enlarging community sequences. The proposed optimization strategies significantly reduce the spatiotemporal cost of the two-phase algorithm.