Closed-Form Leonardo-Type Sequences in Tribonacci, Third-Order Pell and Padovan Numbers

In this paper, we derive unified closed-form solutions for third-order nonhomogeneous linear recurrence relations of Leonardo-type sequences, where the input term is a polynomial. By decomposing each recurrence into homogeneous and particular components, we establish explicit formulas that depend jointly on the root multiplicity of the characteristic equation and the degree of the input polynomial. Resonance effects, which occur when the input polynomial interacts with repeated roots, are discussed in the general framework; however, in our illustrative examples the case r = 0 is considered, meaning all three roots of the characteristic equation are distinct from 1. In this setting, classical sequences such as Tribonacci, Pell, and Padovan numbers emerge naturally as the homogeneous counterparts of generalized Leonardo-type recurrences. These examples illustrate how closed-form expressions clarify the interaction between characteristic roots, polynomial inputs, and resonance phenomena, while also providing templates for applications across discrete mathematics, combinatorics, computational number theory, algorithm analysis, cryptographic constructions, and discrete models in physics and biology, demonstrating the broader impact of recurrence theory. Our framework provides complete closed-form expressions for all cases studied, thereby extending the classical theory of recurrence relations and advancing the analysis of higher-order integer sequences.