I heard a talk at Indiana University last March by Timothy Chow. Here's his abstract, which seems to give a negative answer to your question about rectangles whose sides have non-integer ratio:
It is a classical result that if k is a positive integer, then the number of lattice paths from (0,0) to (a+1,b) taking unit north or east steps that avoid touching or crossing the line x = ky is
(a+b choose b) - k (a+b choose b-1).
Disappointingly, no such simple formula is known if k is rational but not an integer (although there does exist a determinant formula). We show that if we replace the straight-line boundary with a periodic staircase boundary, and if we choose our starting and ending points carefully, then the natural generalization of the above simple formula holds. By varying the boundary slightly we obtain other cases with simple formulas, but it remains somewhat mysterious exactly when a simple formula can be expected. Time permitting, we will also describe some recent related work by Irving and Rattan that provides an alternative proof of some of our results.
This is joint work with Chapman, Khetan, Moulton, and Waters.
