Symmetry Properties and Chaotic Dynamics of Trisenne–Lucas Numbers

The main objective of this paper is to introduce a new class of sequences called the Trisenne–Lucas numbers {m_n^(3)}n∈N​. These sequences extend classical Lucas-type structures and provide a new framework for studying discrete dynamical systems and special number sequences. We begin by establishing their recurrence relation and investigate the existence and uniqueness of the sequence generated by this relation. Based on this construction, we derive several fundamental properties, including Binet’s formula and the generating function, which allow an explicit representation of the terms of the sequence.

Furthermore, we study the associated symmetric functions and explore their role in describing algebraic structures underlying the sequence. Explicit formulas and a variety of summation identities involving the Trisenne–Lucas numbers are also presented, providing deeper insight into their combinatorial behavior. In addition, we examine products of Trisenne–Lucas numbers with certain second-order sequences, revealing interactions between different recursive structures.

Finally, we investigate the dynamical aspects of the Trisenne–Lucas map. In particular, we analyze its chaotic behavior using bifurcation diagrams and Lyapunov exponents, highlighting rich nonlinear dynamics and sensitivity to parameter variations. These results extend classical studies of Mersenne- and Lucas-type sequences and reveal new connections between discrete mathematics, symmetric function theory, and dynamical systems. The proposed framework opens new directions for further research in both algebraic structures and chaos theory.