Sparse, sequential patterns of neural activity have been observed in numerous brain areas during timekeeping and motor sequence tasks. Inspired by such observations, we construct a model of the striatum, an all-inhibitory circuit where sequential activity patterns are prominent, addressing the following key challenges: (i) obtaining control over temporal rescaling of the sequence speed, with the ability to generalize to new speeds; (ii) facilitating flexible expression of distinct sequences via selective activation, concatenation, and recycling of specific subsequences; and (iii) enabling the biologically plausible learning of sequences, consistent with the decoupling of learning and execution suggested by lesion studies showing that cortical circuits are necessary for learning, but that subcortical circuits are sufficient to drive learned behaviors. The same mechanisms that we describe can also be applied to circuits with both excitatory and inhibitory populations, and hence may underlie general features of sequential neural activity pattern generation in the brain.