Level algebras of type 2
  In this paper we study standard graded artinian level algebras, in particular those whose socle-vector has type 2. Our main results are: the characterization of the level $h$-vectors of the form $(1,r,...,r,2)$ for $r\leq 4$; the characterization of the minimal free resolutions associated to each of the $h$-vectors above when $r=3$; a sharp upper-bound (under certain mild hypotheses) for the level $h$-vectors $(1,r,...,a,2)$ of arbitrary codimension $r$ and type 2, which depends on the next to last entry $a$. 
Fabrizio Zanello