Infinite Abelian group extracted from an infinite sequence

Abstract

In this paper, we introduce a reversed symmetric Tribonacchi sequence and establish a new recurrence relation associated with it. We construct an infinite series involving binomial coefficients derived from the classical Tribonacchi sequence, leading to the formulation of an infinite Abelian group. Furthermore, we develop a set of 2 by 2 matrices, forming a matrix subgroup by employing the concepts of eigenvalues and eigenvectors tied to the reversed symmetric Tribonacchi sequence. Our results include closed-form expressions and combinatorial representations for the sums of terms in these newly defined sequences. Finally, we explore the interrelationships among these sequences, demonstrating how they naturally give rise to algebraic group and subgroup structures.