The Generalized Order- k Lucas Sequences in Finite Groups

Abstract

We study the generalized order-$k$ Lucas sequences modulo $m$. Also, we define the $i$th generalized order-$k$ Lucas orbit $l_A^{i,\left\{{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_{k-1}\right\}}$($G$) with respect to the generating set $A$ and the constants ${\alpha }_1,{\alpha }_2$, and ${\alpha }_{k-1}$ for a finite group $G=\langle A\rangle$. Then, we obtain the lengths of the periods of the $i$th generalized order-$k$ Lucas orbits of the binary polyhedral groups $\langle n,2,2\rangle ,$ $\langle 2,n,2\rangle ,$ $\langle 2,2,n\rangle$ and the polyhedral groups $\left(n,2,2\right),\left(2,n,2\right),\left(2,2,n\right)$ for $1\le i\le k$.