Asian Journal of Mathematics and Computer Research

Recurrence sequences are widely used mathematical models with applications across many disciplines. Beyond classical second-order sequences, higher-order families such as Tribonacci, Tetranacci, and Pentanacci reveal richer algebraic structures, with the study of their characteristic roots and recurrence relations advancing symbolic recurrence theory.

In this paper, we present unified closed-form solutions for third-order nonhomogeneous linear recurrence relations of Leonardo-type sequences, where the input term is taken as a polynomial. By decomposing each recurrence into homogeneous and particular components, we obtain explicit formulas that depend jointly on the multiplicity of the characteristic roots and the degree of the input polynomial. Although the general framework accounts for resonance phenomena arising from repeated roots, our illustrative examples focus on the non-resonant case r = 0, where all three roots of the characteristic equation are distinct from 1.

Within this setting, we investigate several notable families of generalized Tribonacci numbers, which appear as homogeneous analogues of the original nonhomogeneous relations. Classical sequences such as the adjusted Pell-Padovan, third-order Lucas-Pell, third-order Fibonacci-Pell, Pell-Perrin, Pell-Padovan, adjusted Jacobsthal-Padovan, Jacobsthal-Perrin (Jacobsthal-Perrin-Lucas), Jacobsthal-Padovan, Narayana, and Narayana-Lucas numbers arise naturally as special cases of the Leonardo-type framework. These examples illustrate how closed-form expressions clarify the interaction between characteristic roots, polynomial inputs, and resonance effects, while also providing templates for applications in discrete mathematics, combinatorics, computational number theory, algorithmic analysis, cryptography, and discrete models in physics and biology. A further illustration is given by considering the case where the input polynomial has degree s = 3, which serves as a natural extension of the classical situations with s = 0. Under identical initial conditions, the homogeneous dynamics reproduce the well-known Pell-Padovan, Lucas-Pell, Fibonacci-Pell, Pell-Perrin, and Pell-Padovan sequences, while the cubic input enriches the particular solution. This demonstrates the continuity of the framework across polynomial degrees and emphasizes the role of initial values in shaping the resulting closed forms.

Beyond their theoretical contribution, the explicit constructions offer pedagogical value by enabling students to engage directly with nonhomogeneous recurrences through accessible formulas rather than lengthy computations. Thus, the study demonstrates both the novelty and interdisciplinary reach of generalized Leonardo-type sequences, furnishing researchers with extended tools for higher-order recurrence analysis and educators with clear examples for teaching advanced recurrence methods.