Fano 3-folds with divisible anticanonical class

We show the nonvanishing of H-0(X,- K-X) for any a Fano 3-fold X for which - KX is a multiple of another Weil divisor in Cl(X). The main case we study is Fano 3-folds with Fano index 2: that is, 3-folds X with rank Pic(X) = 1, Q-factorial terminal singularities and -K-X = 2A for an ample Weil divisor A. We give a first classification of all possible Hilbert series of such polarised varieties (X, A) and deduce both the nonvanishing of H-0(X,- K-X) and the sharp bound (-K-X)(3) >= 8/165. We find the families that can be realised in codimension up to 4.