Simple formulas for lattice paths avoiding certain periodic staircase boundaries

There is a strikingly simple classical formula for the number of lattice paths avoiding the line x = k y when k is a positive integer. We show that the natural generalization of this simple formula continues to hold when the line x = k y is replaced by certain periodic staircase boundaries—but only under special conditions. The simple formula fails in general, and it remains an open question to what extent our results can be further generalized.