The Tribonacci-type balancing numbers and their applications. (English) Zbl 07804646
Summary: In 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 \((2,2,n)\), \((2,n,2)\), \((n,2,2)\), \((2,3,3)\), \((2,3,4)\), \((2,3,5)\).
MSC:
| 11K31 | Special sequences |
| 39B32 | Functional equations for complex functions |
| 11B50 | Sequences (mod \(m\)) |
| 20F05 | Generators, relations, and presentations of groups |
| 11C20 | Matrices, determinants in number theory |
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