  We consider a vertex reinforced random walk on the integer lattice with
sub-linear reinforcement. Under some assumptions on the regular variation of
the weight function, we characterize whether the walk gets stuck on a finite
interval. When this happens, we estimate the size of the localization set. In
particular, we show that, for any odd number $N$ larger than or equal to 5,
there exists a vertex reinforced random walk which localizes with positive
probability on exactly $N$ consecutive sites.
