Rational Dyck Paths

Given a positive rational q, we consider Dyck paths of height at most two, subject to constraints on the number of consecutive peaks and consecutive valleys depending on q. We introduce a general class of Dyck paths, named rational Dyck paths, and provide the associated generating function based on their semilength, along with a construction for this class. Moreover, we characterize certain subsets of rational Dyck paths that are enumerated by the Q-bonacci numbers.