In previous works, some relations between Pringsheim continued fractions and vertices of the paths of minimallength on the suborbital graphs Fu,N were investigated. Then, for special vertices, the relations between thesevertices and Fibonacci numbers were examined. On the other hand, Koshy studied relation between recurrencerelations of Fibonacci numbers, Pell numbers and generating functions. In this work, it is showed that every vertex on the path of minimal length of suborbital graph Fu,N has a Pringsheim continued fraction. Then, by Koshy’s motivation, the generating function of the recurrence relation of these Pringsheim continued fractions are examined.