Exact solution of two classes of prudent polygons

Prudent walks are self-avoiding walks on a lattice which never step into the direction of an already occupied vertex. We study the closed version of these walks, called prudent polygons, where the last vertex of the walk is adjacent to its first one. More precisely, we give the half-perimeter generating functions of two subclasses of prudent polygons on the square lattice, which turn out to be algebraic and non-D-finite, respectively.