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 -K_X 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.