Square lattice walks avoiding a quadrant

In the past decade, a lot of attention has been devoted to the enumeration of walks with prescribed steps confined to a convex cone. In two dimensions, this means counting walks in the first quadrant of the plane (possibly after a linear transformation).But what about walks in non-convex cones? We investigate the two most natural cases: first, square lattice walks avoiding the negative quadrant Q1={(i,j):i<0 and j<0}Q1={(i,j):i<0 and j<0}, and then, square lattice walks avoiding the West quadrant Q2={(i,j):i