  We consider the generating function of the algebraic area of lattice walks,
evaluated at a root of unity, and its relation to the Hofstadter model. In
particular, we obtain an expression for the generating function of the n-th
moments of the Hofstadter Hamiltonian in terms of a complete elliptic integral,
evaluated at a rational function. This in turn gives us both exact and
asymptotic formulas for these moments.
