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September 5, 2025Open Access

On the generalization of third order linear recurrence relations and matrix representation

Key Points

The paper establishes a matrix representation of third order linear recurrence relations to generate Tribonacci numbers.
It shows that generalizing these relations provides a framework for reproducing previous outcomes under new conditions.
Initial terms and coefficients can be adjusted to demonstrate secure message communication through varying parameters.
A numerical example illustrates both the forward and retrieval processes of messages based on the proposed model.

Abstract

In this paper, generalization {V₍} ({V₈}, {p₈}) =₉=₁^3{{p₁}{V₍-₉}}, j=1, 2, 3, n 4 of the third order linear recurrence relations is considered and represented in two different ways to generate Tribonacci numbers. Thereby, a matrix representation is established to engender numbers and to investigate identities and results in the generalized form. It is observed from the obtained generalized results that relevant previous outcomes become the special cases. On picking up {V₉, p₉}, j=1, 2, 3 initial terms, coefficients of the linear recurrence relations and the power n of the matrix{M^n} arbitrarily, to demonstrate the communication of message. For secruty, same message can be send frequently with a different paprmetres by varying {V₉, p₉}, j=1, 2, 3 and n. A numerical example is presented to illustrate to forward and retrieve the messages.

On the generalization of third order linear recurrence relations and matrix representation

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