A new approach to Olivier sequences

Abstract

In this study, we define the k-Olivier, k-Olivier-Lucas, and Modified k-Olivier sequences and give some terms of these sequences. Then, we obtain the generating functions, summation formulas, etc. Also, we obtain the Binet formulas in three different ways. The first is in the known classical way, the second is with the help of the sequence's generating functions, and the third is with the help of the matrices. In addition, the relations between the terms of the k-Olivier, k-Olivier-Lucas, Modified k-Olivier, third-order Fibonacci-Pell, Adjusted Pell-Padovan, Olivier, Olivier-Lucas, Modified Olivier, third-order Lucas-Pell, Pell-Padovan, and Pell-Perrin sequences are presented. Finally, we associate the terms of these sequences with matrices.