On generalized (k, r)-Pell and (k, r)-Pell–Lucas numbers

Bahar Kuloğlu and Engin Özkan
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 28, 2022, Number 4, Pages 765–777
DOI: 10.7546/nntdm.2022.28.4.765-777
Full paper (PDF, 273 Kb)

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Authors and affiliations

Bahar Kuloğlu
Department of Mathematics, Faculty of Arts and Sciences,
Erzincan Binali Yıldırım University, Erzincan, Türkiye

Engin Özkan
Department of Mathematics, Faculty of Arts and Sciences,
Erzincan Binali Yıldırım University, Erzincan, Türkiye

Abstract

We introduce new kinds of -Pell and -Pell–Lucas numbers related to the distance between numbers by a recurrence relation and show their relation to the -Pell and -Pell–Lucas numbers. These sequences differ both according to the value of the natural number and the value of a new parameter in the definition of this distance. We give several properties of these sequences. In addition, we establish the generating functions, some important identities, as well as the sum of the terms of the generalized -Pell and -Pell–Lucas numbers. Furthermore, we indicate another way to obtain the generalized -Pell and -Pell–Lucas sequences from the generating function, in connection to graphs.

Keywords

• Generalizations of Pell numbers
• k-Pell numbers
• r-distance Pell numbers
• Graphs
• Generating functions

2020 Mathematics Subject Classification

• 11B39
• 11B83
• 05A15

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Manuscript history

• Received: 12 June 2022
• Revised: 24 October 2022
• Accepted: 29 November 2022
• Online First: 30 November 2022