Explicit Formulas for Polynomial-type Particular Solutions of Generalized Leonardo-type Sequences

Recurrence relations provide a powerful framework for understanding numerical sequences, with applications spanning both classical and modern mathematics. In earlier work, explicit iterative formulas for polynomial-type particular solutions of generalized Leonardo-type sequences were derived. The present article complements that theoretical foundation by presenting detailed examples for orders m = 1, 2, 3, 4, where the input function C(n) = p(n) is a polynomial in n. For such sequences, we construct particular solutions of the form \(W_n^{(C)}=n^r\left(\sum_{i=0}^s A_i n^i\right)\) and provide explicit iterative formulas for the coefficients Ai. These examples illustrate how the multiplicity r of the root 1 in the characteristic equation influences the structure of the solution and highlights resonance phenomena in non-homogeneous settings. By focusing on representative low-order recurrences, the paper demonstrates how the general framework specializes to concrete cases, thereby confirming its consistency with classical identities and clarifying the corrective terms that arise under resonance. In this way, the study provides both theoretical validation and pedagogical clarity, bridging abstract recurrence theory with accessible symbolic computation and practical modeling. Beyond their theoretical contribution, the results presented here demonstrate broad applicability across diverse fields. Recurrence sequences naturally arise in contexts ranging from physics and engineering to biology, computer science, and even artistic design. In each of these domains, they provide a symbolic language for describing growth, resonance, feedback, and complexity. We present three representative examples that illustrate how resonance and root multiplicities shape the construction of particular solutions in polynomial-driven recurrence relations. In the generalized Fibonacci case, the input polynomial is non-resonant, and the particular solution can be obtained directlywithout correction. The generalized Mersenne case highlights resonance, where the root 1 of the characteristic equation requires a multiplicity-aware adjustment. The generalized Avicenna case demonstrates higher-order resonance, with root 1 of multiplicity four, showing how corrective factors grow in complexity as multiplicity increases. Together, these three examples confirm the robustness of the iterative framework, clarify the algebraic structure of resonance-aware corrections, and highlight the adaptability of the method across different orders. They also serve as pedagogical anchors, transforming abstract iteration into accessible computations, and as templates for interdisciplinary modeling. Building on these representative cases, the general theorem provides a systematic procedure for deriving polynomial-type particular solutions of generalized Leonardo-type sequences, valid for arbitrary order and input polynomials. In this way, the theorem unifies the case-by-case constructions into a single iterative scheme, reinforcing both the theoretical depth and the practical utility of the study.