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 πk(m)πk(m). It is proved that for every odd number k  , πk(k2+4)=4(k2+4)πk(k2+4)=4(k2+4).