Generalized Jacobsthal-Narayana Numbers and Generalized Co-Jacobsthal-Narayana Numbers

This paper introduces two innovative third-order recurrence sequences: the generalized Jacobsthal-Narayana sequence and the co-Jacobsthal-Narayana sequence. It examines their interrelated properties, including Binet’s formulas, generating functions, Simson’s formulas, and matrix representations, as well as their special subsequences. The study highlights unique relationships between recurrence equations and roots of characteristic equations, uncovering novel properties. These sequences hold promising potential for applications in various fields, such as number theory, combinatorics, cryptography, and modeling phenomena in physics, biology, and economics. For instance, the matrix representation of co-Jacobsthal-Narayana-Lucas numbers has implications for cryptographic systems.