Authors: Mireille Bousquet-Mélou ¹

• 1 Laboratoire Bordelais de Recherche en Informatique

A self-avoiding walk on the square lattice is $\textit{prudent}$, if it never takes a step towards a vertex it has already visited. Préa was the first to address the enumeration of these walks, in 1997. For 4 natural classes of prudent walks, he wrote a system of recurrence relations, involving the length of the walks and some additional "catalytic'' parameters. The generating function of the first class is easily seen to be rational. The second class was proved to have an algebraic (quadratic) generating function by Duchi (FPSAC'05). Here, we solve exactly the third class, which turns out to be much more complex: its generating function is not algebraic, nor even $D$-finite. The fourth class ―- general prudent walks ―- still defeats us. However, we design an isotropic family of prudent walks on the triangular lattice, which we count exactly. Again, the generating function is proved to be non-$D$-finite. We also study the end-to-end distance of these walks and provide random generation procedures.