The complex-type cyclic-Fibonacci sequence and its applications

In the present paper, we aim to generalize the notion of complex-type Fibonacci sequences to complex-type cyclic Fibonacci sequences. Firstly, we define the complex-type cyclic-Fibonacci sequence and then we give miscellaneous  properties of this sequence by using the matrix method. Also, we study the complex-type cyclic-Fibonacci sequence modulo $m$. In addition, we describe the complex-type cyclic-Fibonacci sequence in a $2$-generator group and investigate that in finite groups in details. Then, as our last result, we obtain the lengths of the periods of the complex-type cyclic-Fibonacci sequences in dihedral groups $D_{2}$, $D_{3}$, $D_{4}$, $D_{5}$, $D_{6}$ and $D_{8}$ with respect to their generating sets.