Algebraic properties of finite simple groups via the (r,k)-bonacci sequence and more

The main goal of this paper is to determine algebraic properties of finite groups via the [Formula: see text]-bonacci sequences. First, we give the relationships between the [Formula: see text]-bonacci numbers, the Fibonacci-[Formula: see text] numbers, the Narayana numbers, and the [Formula: see text]-generalized order-[Formula: see text] Perrin numbers. Then, we study the [Formula: see text]-bonacci numbers modulo [Formula: see text] and give some results concerning the periods of the [Formula: see text]-bonacci sequences for any [Formula: see text] and [Formula: see text]. In addition, we extend the [Formula: see text]-bonacci sequences to [Formula: see text]-generator groups and investigate these sequences in finite groups. Finally, we obtain the periods of the [Formula: see text]-bonacci sequences in well-known classes of finite groups as an application of the results obtained.