Asian Journal of Pure and Applied Mathematics

This paper studies second-order nonhomogeneous linear recurrence relations of Leonardo type in which the forcing term is a polynomial. The main aim is to present closed-form expressions by decomposing each sequence into its homogeneous part and a polynomial particular solution. The method uses the roots of the associated characteristic equation and organises the solution according to the multiplicity of the root 1. For the non-resonant case needed in the examples, the coefficients of the polynomial particular solution are obtained through an explicit iterative scheme. The resulting formula is then applied to two classical recurrence families. First, the parameter choice a₁ = 6 and a₂ = −1 yields the generalized balancing family, including the balancing, modified Lucas-balancing, and Lucas-balancing numbers. Closed-form solutions are recorded for polynomial inputs of degrees 0 through 7 in these cases. Secondly, the parameter choice a₁ = 1 and a₂ = −\(\frac{1}{4}\) gives the generalized Oresme family, for which modified Oresme and Oresme numbers are treated under polynomial inputs of degrees 0 through 3. The repeated Oresme characteristic root \(\frac{1}{2}\) does not create resonance because the classification depends only on the occurrence of the root 1. The results provide a unified presentation of polynomially forced Leonardo-type recurrences and their homogeneous counterparts while retaining the classical balancing and Oresme sequences as special cases.