Journal of Advances in Mathematics and Computer Science

The objective of this study is to derive explicit closed-form solutions for second-order nonhomogeneous linear recurrence relations with polynomial inputs, formulated in terms of generalized Fibonacci-type and generalized Pell-type sequences. A central aspect of the framework is the restriction to the non-resonant case r = 0, which ensures that the characteristic equation admits two distinct roots, neither equal to unity. This setting avoids resonance phenomena and allows particular solutions to be constructed in their simplest form.

Within this framework, we highlight two principal families: the generalized Fibonacci sequences, corresponding to (a₁,a₂) = (1,1) with characteristic roots {\(\frac{1+\sqrt{5}}{2}\), \(\frac{1-\sqrt{5}}{2}\)}, and the generalized Pell sequences, corresponding to (a₁,a₂) = (2,1) with roots {1 + \(\sqrt{2}\), 1 - \(\sqrt{2}\)}. In both cases, closed-form solutions are obtained under polynomial inputs of degrees s = 0,1,2,3,4,5,6,7, covering constant through septic forcing terms. These results clarify how the polynomial degree shapes the explicit formulas, while the homogeneous counterparts (classical Fibonacci, Lucas, Pell, and Pell–Lucas sequences) emerge naturally when the input polynomial is suppressed.

The study thus provides a unified framework that connects these classical integer sequences with their nonhomogeneous extensions, offering closed forms that are both theoretically significant and pedagogically accessible. By focusing exclusively on the non-resonant case r = 0, the exposition remains streamlined while still encompassing a broad range of polynomial inputs and sequence families.

Beyond their theoretical contribution, the explicit examples provide pedagogical value by allowing students to engage directly with nonhomogeneous recurrences under polynomial inputs without excessive computation. Thus, the study demonstrates both the novelty and interdisciplinary impact of generalized Fibonacci and Pell sequences in the nonhomogeneous setting. The explicit examples serve as accessible templates for teaching advanced recurrence methods, making the paper valuable not only to researchers but also to educators and students, while also providing classroom illustrations of polynomial inputs of varying degrees.

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 Leonardo-type recurrences.