We have proved that, for the 3-step Fibonacci recurrence and any finite p-group of exponent p and nilpotency class 3, the length of a fundamental period of any loop satisfying the recurrence must divide the period of the ordinary 3-step Fibonacci sequence in the field GF(p).
DİKİCİ, Ramazan and SMITH, Geoff C. (1997) "Fibonacci Sequences in Finite Nilpotent Groups," Turkish Journal of Mathematics: Vol. 21: No. 2, Article 1. Available at: https://journals.tubitak.gov.tr/math/vol21/iss2/1