The Tribonacci-type balancing numbers and their applications
Mathematica Moravica, Tome 27 (2023) no. 1.

Voir la notice de l'article dans eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts

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 $(2,2,n)$, $(2,n,2)$, $(n,2,2)$, $(2,3,3)$, $(2,3,4)$, $(2,3,5)$.
Mots-clés : The Tribonacci-type balancing sequence, Matrix, Group, Presentation, Period, Rank. 
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Sakine Hulku; Ömür Deveci. The Tribonacci-type balancing numbers and their applications. Mathematica Moravica, Tome 27 (2023) no. 1. https://geodesic-test.mathdoc.fr/item/MM3_2023_27_1_a2/