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In this article, we consider compositions of positive integers with 2s and 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 2s and 3s related 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 2s and 3s.

In this paper, we define the quaternion Padovan sequence. We obtain some identities for this quaternion sequence. Also, we derive a plastic-like ratio. [ABSTRACT FROM AUTHOR]

Spinors are components of a complex vector space that can be related to Euclidean space in both geometry and physics. In essence, the forms of usage include quaternions that are equivalent to Pauli spin matrices, which may be produced by thinking of a quaternion matrix as the compound. This study's objective is the spinor structure that forms based on the quaternion algebra. In this work, first, spinors have been mathematically presented. Then, Padovan and Perrin spinors have been defined using the Padovan and Perrin quaternions. Later, we defined the algebraic structure for these spinors. Finally, we have established certain identities such as the Binet formulas and generating functions for Padovan and Perrin spinors. [ABSTRACT FROM AUTHOR]

In this study, we define a new generalization of the Padovan numbers, which shall also be called the bi-periodic Padovan sequence. Also, we consider a generalized bi-periodic Padovan matrix sequence. Finally, we investigate the Binet formulas, generating functions, series and partial sum formulas for these sequences. [ABSTRACT FROM AUTHOR]

In the present work, two new recurrences of the Jacobsthal sequence are defined. Some identities of these sequences which we call the Jacobsthal array is examined. Also, the generating and series functions of the Jacobsthal array are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Fibonacci and Padovan sequences. We introduce the quinary Fibonacci-Padovan sequences whose compounds are the Fibonacci and Padovan sequences. We derive the Binet-like formulas, the generating functions and exponential generating functions of these sequences. Also, we obtain some binomial identities, series and sums for them. [ABSTRACT FROM AUTHOR]

In the present work we introduce Bernoulli-Padovan numbers and polynomials. We give their generating functions of the Bernoulli-Padovan numbers and polynomials. We establish various relations involving the Bernoulli-Padovan numbers and polynomials by considering the Pado-derivative. We describe Pado-Bernoulli matrices in terms of the Bernoulli-Padovan numbers and polynomials. We establish a factorisation of the Pado-Bernoulli matrix by using a generalised Pado-Pascal matrix, and obtain the inverse of the Pado-Bernoulli matrix. Also, we give a relationship between the Pado-Bernoulli matrix and the Pado-Pascal matrix. [ABSTRACT FROM AUTHOR]

In the present work, two new recurrences of the Padovan sequence given with delayed initial conditions are defined. Some identities of these sequences which we call the Padovan arrays were examined. Also, generating and series functions of the Padovan arrays are examined. [ABSTRACT FROM AUTHOR]

In this study, one-dimensional, two-dimensional, three-dimensional and n-dimensional recurrences of the (p; q)-Fibonacci sequence are examined and their some identities are given. [ABSTRACT FROM AUTHOR]

This paper examines the Fibonacci quaternion sequence with quadruple-produce components, and demonstrates a golden-like ratio and some identities for this sequence. Its generating and exponential generating functions are given. Along with these, its series and binomial sum formula are established. [ABSTRACT FROM AUTHOR]

In the present work, we consider the Padovan sequence and define a sequence (called Quadrovan) that is a new generalization. In addition, we give the previously defined Tridovan sequence as a generalization of the Padovan sequence. We derive the Binet-like formulas, the generating functions and the exponential generating functions for the Tridovan and Quadrovan sequences. Also, we establish their series and matrices. [ABSTRACT FROM AUTHOR]

In this paper we first introduce a class of (s, t)-Padovan and (s, t)-Perrin quaternions which generalizes Padovan and Perrin quaternions, and then we derive new Binet-like formulas, generating functions and certain binomial sums for these quaternions. [ABSTRACT FROM AUTHOR]