  We present an {\it ab initio} calculation of small numbers of trapped,
strongly interacting fermions using the Green's Function Monte Carlo method
(GFMC). The ground state energy, density profile and pairing gap are calculated
for particle numbers $N = 2 \sim 22$ using the parameter-free "unitary"
interaction. Trial wave functions are taken of the form of correlated pairs in
a harmonic oscillator basis. We find that the lowest energies are obtained with
a minimum explicit pair correlation beyond that needed to exploit the
degeneracy of oscillator states. We find that energies can be well fitted by
the expression $a_{TF} E_{TF} + \Delta {\rm mod}(N,2)$ where $E_{TF}$ is the
Thomas-Fermi energy of a noninteracting gas in the trap and $\Delta$ is a
pairing gap. There is no evidence of a shell correction energy in the
systematics, but the density distributions show pronounced shell effects. We
find the value $\Delta= 0.7\pm 0.2\omega$ for the pairing gap. This is smaller
than the value found for the uniform gas at a density corresponding to the
central density of the trapped gas.
