  Our studies are related to a special class of FASS-curves, which can be
described in a node-rewriting Lindenmayer-system. These ortho-tile (or
diagonal) type recursive curves inducing Hamiltonian paths. We define a special
directed graph on a rectangular grid, and we enumerate all Hamiltonian paths on
this graph. Our formulas are strongly related to both the Fibonacci numbers and
the domino tilings of chessboards. The constructability of the regular $17$-gon
with straightedge and compass is also related.
