  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.
