Using large-scale quantum Monte Carlo simulations we study bosons hopping on a triangular lattice with nearest (V) and next-nearest (V) neighbor repulsive interactions. In the limit where V=0 but V is large, we find an example of an unusual period-three striped supersolid state that is stable at 1/2-filling. We discuss the relationship of this state to others on the rich ground-state phase diagram, which include a previously-discovered nearest-neighbor supersolid, a uniform superfluid, as well as several Mott insulating phases. We study several superfluid- and supersolid-to-Mott phase transitions, including one proposed by a recent phenomenological dual vortex field theory as a candidate for an exotic deconfined quantum critical point. We find no examples of unconventional quantum criticality among any of the interesting phase transitions in the model.
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