Asian Basic and Applied Research Journal

This paper develops unified closed-form formulas for third-order nonhomogeneous linear recurrence relations of Leonardotype sequences, where the driving term is taken to be a polynomial. The method proceeds by decomposing each recurrence into its homogeneous and particular components, yielding explicit solutions that depend simultaneously on the multiplicity of the characteristic roots and the degree of the input polynomial. While the general framework accommodates resonance phenomena arising from repeated roots, the examples presented here concentrate on the non-resonant case r = 0, in which all three roots of the characteristic equation are distinct from 1.

Within this setting, several notable families are examined, including the generalized third-order Jacobsthal, generalized Graham, and generalized Jacobsthal–Narayana sequences. Classical counterparts such as the third-order Jacobsthal, modified Jacobsthal, Jacobsthal–Lucas, Graham, Graham–Lucas, Jacobsthal–Narayana, and Jacobsthal–Narayana–Lucas numbers appear naturally as specializations of the Leonardo-type framework. These cases 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 symbolic modeling in physics and biology.

A further illustration is obtained by fixing specific coefficients ci in the input polynomial and considering degrees s = 0, 1, 2, 3. Under identical recurrence parameters, the homogeneous dynamics reproduce the well-known third-order Jacobsthal, modified Jacobsthal, and Jacobsthal–Lucas sequences, while the constant, linear, quadratic, and cubic inputs enrich the particular solutions. 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 also serve pedagogical purposes by enabling students to engage directly with nonhomogeneous recurrences through accessible formulas rather than lengthy computations. At the same time, they furnish researchers with extended tools for higher-order recurrence analysis. In this way, the study highlights both the novelty and interdisciplinary reach of generalized Leonardo-type sequences, strengthening recurrence theory while supporting applications across mathematics, computer science, engineering, and the natural sciences.