  We study the enumeration of closed walks of given length and algebraic area
on the honeycomb lattice. Using an irreducible operator realization of
honeycomb lattice moves, we map the problem to a Hofstadter-like Hamiltonian
and show that the generating function of closed walks maps to the grand
partition function of a system of particles with exclusion statistics of order
$g=2$ and an appropriate spectrum, along the lines of a connection previously
established by two of the authors. Reinterpreting the results in terms of the
standard Hofstadter spectrum calls for a mixture of $g=1$ (fermion) and $g=2$
exclusion whose physical meaning and properties require further elucidation. In
this context we also obtain some unexpected Fibonacci sequences within the
weights of the combinatorial factors appearing in the counting of walks.
