Generalized Olivier Numbers
Yüksel Soykan *
Department of Mathematics, Faculty of Science, Zonguldak Bulent Ecevit University, Zonguldak -67100, Turkey.
*Author to whom correspondence should be addressed.
Abstract
In this paper, we introduce and investigate the generalized Olivier sequences and we deal with, in detail, two special cases, namely, Olivier and Olivier-Lucas sequences. We present Binet's formulas, generating functions, Simson formulas, and the summation formulas for these sequences. We also provide various
matrices and identities associated with these sequences. Furthermore, we show that there are close relations between Olivier, Olivier-Lucas and adjusted Pell-Padovan, third order Lucas-Pell, third order Fibonacci-Pell, Pell-Perrin, Pell-Padovan numbers. Moreover, we give some identities and matrices related with these
sequences.
Keywords: Olivier numbers, Olivier-Lucas numbers, Pell-Padovan numbers, Pell-Perrin numbers, third order Fibonacci-Pell numbers, third order Lucas-Pell numbers numbers, Fibonacci numbers, Lucas numbers