  We introduce a strongly correlated mean-field theory of the ionic Hubbard
model on the triangular lattice with alternating stripes of site energy using
Barnes-Coleman slave bosons. We study the paramagnetic phases of this theory at
three quarters filling, where it is a model of Na_0.5CoO_2, Rb_0.5CoO_2, and
K_0.5CoO_2. This theory has two bands of fermionic quasi-particles: one of
which is filled or nearly filled and hence weakly correlated; the other is
half-filled or nearly half-filled and hence strongly correlated. Further
results depend strongly on the sign of the hopping integral, t. The light band
is always filled for t>0, but only becomes filled for |Delta/t|>=1.5 for t<0,
where Delta is the difference in the site energies of the two sublattices. A
metal--charge transfer insulator transition occurs at |Delta/t|=5.0 for t>0 and
|Delta/t|=8.0 for t<0. In the charge transfer insulator complete charge
disproportionation occurs: one sublattice is filled and the other in half
filled. We compare our results with exact diagonalisation calculations and
experiments on Na_0.5CoO_2, and discuss the relevance of our results to
Rb_0.5CoO_2 and K_0.5CoO_2. In particular we propose a resolution of seemingly
contradictory experimental results on Na_0.5CoO_2. Many experiments suggest
that there is a charge gap, yet quantum oscillations are observed suggesting
the existence of quasiparticle states at arbitrarily low excitation energies.
We argue that the heavy band is gapped while the light band, which contains
less than 1 charge carrier per 100 unit cells, remains ungapped.
