The Tribonacci-type balancing numbers and their applications

Abstract

N this paper, we define the Tribonacci-type balancing numbers via a Diophantine equation with a complex variable and then give their miscellaneous properties. Also, we study the Tribonacci-type balancing sequence modulo m and then obtain some interesting results concerning the periods of the Tribonacci-type balancing sequences for any m. Furthermore, we produce the cyclic groups using the multiplicative orders of the generating matrices of the Tribonacci-type balancing numbers when read modulo m. Then give the connections between the periods of the Tribonacci-type balancing sequences modulo m and the orders of the cyclic groups produced. Finally, we expand the Tribonacci-type balancing sequences to groups and give the definition of the Tribonacci-type balancing sequences in the 3-generator groups and also, investigate these sequences in the non-abelian finite groups in detail. In addition, we obtain the periods of the Tribonacci-type balancing sequences in the polyhedral groups (2, 2, n), (2, n, 2), (n, 2, 2), (2, 3, 3), (2, 3, 4), (2, 3, 5).