  Spinless, interacting electrons on a finite size triangular lattice moving in
an extremely strong perpendicular magnetic field are studied and compared with
the results on a square lattice. Using a Falicov-Kimball model, the effects of
the magnetic field, Coulomb correlation and finite system size on their energy
spectrum are observed. It is possible to induce a gap in the spectrum by tuning
the magnetic field even in the absence of correlation, though extra states
appear in the gap due to finite size. An orbital current is calculated for both
the square and triangular lattice with and without electron correlation. In the
noninteracting limit, the bulk current shows several patterns, while the edge
current shows oscillations with magnetic flux. The oscillations persist in the
interacting limit for the square lattice but not for the triangular lattice.
Using exact diagonalization techniques, the recursive structure of Hofstadter
spectrum is examined with strong electronic correlations for different system
sizes. Electronic correlation is found to suppress these extra states in the
gap in some cases.
