  In the past decade, a lot of attention has been devoted to the enumera-tion
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 Q 1 = {(i, j) : i \textless{} 0 and j \textless{} 0}, and
then, square lattice walks avoiding the West quadrant Q 2 = {(i, j) : i
\textless{} j and i \textless{} --j}. In both cases, the generating function
that counts walks starting from the origin is found to differ from a simple
D-finite series by an algebraic one. We also obtain closed form expressions for
the number of n-step walks ending at certain prescribed endpoints, as a sum of
three hypergeometric terms. One of these terms already appears in the
enumeration of square lattice walks confined to the cone {(i, j) : i +j $\ge$ 0
and j $\ge$ 0}, known as Gessel's walks. In fact, the enumeration of Gessel's
walks follows, by the reflection principle, from the enumeration of walks
starting from (--1, 0) and avoiding Q 1. Their generating function turns out to
be purely algebraic (as the generating function of Gessel's walks). Another
approach to Gessel's walks consists in counting walks that start from (--1, 1)
and avoid the West quadrant Q 2. The associated generating function is D-finite
but transcendental.
