Cycles in the Sequences of Hasse-Allouche

Hasse-Allouche sequences include Collatz-Kakutani sequences. It has previously been shown (2002.0594) with Collatz-Kakutani sequences, generalized to px + q, that the numerical cycles are derived from algebraic cycles. The same is established here with the more complex framework of Hasse-Allouche sequences. The number of elementary functions increases from 2 to r. Again, the beginning and the end of a sequence are connected by a diophantine equation, pm x - rd y - q = 0, where m and d are respectively the numbers of multiplications and divisions. There are still always rotation cycles (q1 q2 ... qm), and derived cycles (x1 x2 ... xm) when qi / (rd - pm) are integers. The enumeration of sub-sequence classes between two integers not divisible by r allows a simple calculation of the q parameter from their rank in the r base.