k-Fibonacci sequences modulo m

We study here the period-length of the k-Fibonacci sequences taken modulo m. The period of such cyclic sequences is know as Pisano period, and the period-length is denoted by pkðmÞ. It is proved that for every odd number k, pkðk2 þ 4Þ ¼ 4ðk2 þ 4Þ.