In this paper, we present a new definition, referred to as the Francois sequence, related to the Lucas-like form of the Leonardo sequence. We also introduce the hyperbolic Leonardo and hyperbolic Francois quaternions. Afterward, we derive the Binet-like formulas and their generating functions. Moreover, we provide some binomial sums, Honsberger-like, d’Ocagne-like, Catalan-like, and Cassini-like identities of the hyperbolic Leonardo quaternions and hyperbolic Francois quaternions that allow an understanding of the quaternions' properties and their relation to the Francois sequence and Leonardo sequence. Finally, considering the results presented in this study, we discuss the need for further research in this field.

In this article, we consider compositions of positive integers with 2sand 3s. We see that these compositions lead us to results that involve Padovan numbers, and we give some tiling models of these compositions. Moreover, we examine some tiling models of the compositions related to the Padovan polynomials and prove some identities using the tiling model’s method. Next, we obtain various identities of the compositions of positive integers with 2sand 3srelated to the Padovan numbers. The number of palindromic compositions of this type is determined, and some numerical arithmetic functions are defined. Finally, we provide a table that compares all of the results obtained from compositions of positive integers with 2sand 3s.

This study explores the formation of polynomials of at most degree $n$ using the first $n+1$ terms of the Jacobsthal and Jacobsthal-Lucas sequences through Lagrange interpolation. The paper provides a detailed examination of the recurrence relations and various identities associated with the Jacobsthal and Jacobsthal-Lucas Lagrange Interpolation Polynomials.