Asian Research Journal of Mathematics

Sequences, both classical and modern in scope, can be analyzed through a versatile framework that remains central to mathematics, namely recurrence relations. Previous investigations established explicit iterative procedures for constructing polynomial–exponential particular solutions of generalized Leonardotype sequences. Building upon that framework, this article develops illustrative examples for the cases m = 3, 4, where the forcing term is given by C(n) = p(n)dⁿ, with p(n) = \(\sum_{i=0}^s c_i n^i\) a polynomial in n. For such recurrences, we derive particular solutions of the form 

\[W_n^{(C)}=n^r\left(\sum_{i=0}^s A_i n^i\right) d^n\]

and demonstrate the computation of the coefficients Ai via the established iterative scheme. These formulas not only provide constructive clarity but also demonstrate how the iterative procedure systematically determines the polynomial part of the solution. The examples reveal how the multiplicity r of the root d in the characteristic polynomial governs the structure of the solution, while resonance phenomena emerge when the forcing term interacts with repeated characteristic roots. Such resonance effects are highlighted in detail, showing their decisive role in shaping the solution’s form and complexity. In addition to the explicit constructions, a brief literature review is included to situate Leonardo-type sequences within their historical development and to highlight recent advances in generalized Leonardotype recurrences. This contextualization underscores the enduring role of recurrence relations in number theory, discrete mathematics, and symbolic computation. By presenting explicit cases, the paper offers a transparent and accessible illustration of the general theory, reinforcing the connection between abstract recurrence analysis and concrete symbolic computation, while also pointing toward potential applications in computational mathematics and combinatorial modeling.