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)**

## Details

### 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

### References

- Aküzüm, Y., & Deveci, Ö. (2020). The Hadamard-type
*k*-step Fibonacci sequences in groups.*Communications in Algebra*, 48, 2844–2856. - Bednarz, U., Włoch A., & Wołowiec-Musial, M. (2013). Distance Fibonacci numbers, their interpretations and matrix generators.
*Commentationes Mathematicae*, 53(1), 35–46. - Belbachir, H., & Bencherif, F. (2011). Unimodality of sequences associated to Pell

Numbers.*Ars Combinatoria*, 102, 305–311. - Bród, D. (2015). On Distance (
*r*,*k*)-Fibonacci Numbers and Their Combinatorial and Graph Interpretations.*Journal of Applied Mathematics*, 2015, Article ID 879510, 1–6. - Bród, D. (2019). On a New One Parameter Generalization of Pell Numbers.
*Annales Mathematicae Silesianae*, 33, 66–76. - Bród, D., Piejko, K., & Włoch, I. (2013). Distance Fibonacci numbers, distance Lucas numbers and their applications.
*Ars Combinatoria*, 112, 397–410. - Catarino, P. (2013). On Some Identities and Generating Functions for k-Pell number.
*International Journal of Mathematical Analysis*, 7, 1877–1884. - Catarino, P., & Vasco, P. (2013). On Some Identities and Generating Functions for
*k*-Pell–Lucas Sequence.*Ars Combinatoria*, 7(98), 4867–4873. - Catarino, P., & Vasco, P. (2017). On Dual
*k*-Pell Quaternions and Octonions.*Mediterranean Journal of Mathematics*, 14, Article ID 75. - Daşdemir, A. (2010). On the Pell, Pell–Lucas and Modified Pell Numbers by Matrix Method.
*Applied Mathematical Sciences*, 5, 3173–3181. - Falcon, S. (2014). Generalized (
*k*,*r*)-Fibonacci Numbers.*General Mathematics Notes*, 25, 148–158. - Falcon, S., & Plaza, A. (2007). On the Fibonacci
*k*-numbers.*Chaos, Solitons & Fractals*, 32, 1615–1624. - Faye, B., & Luca, F. (2015). Pell and Pell–Lucas numbers with only one distinct digit.
*Annales Mathematicae et Informaticae*, 45, 55–60. - Horadam, A. F. (1971). Pell identities.
*The Fibonacci Quarterly*, 9, 245–263. - Horadam, A. F. (1994). Applications of Modified Pell Numbers to Representations.
*Ulam Quarterly*, 3(1), 34–53. - Halıcı, S., & Daşdemir, A. (2010) On some relationships among Pell, Pell–Lucas and modified Pell sequences.
*Sakarya University Journal of Science*, 14(2), 142–145. - Koshy, T. (2014).
*Pell and Pell–Lucas Numbers with Applications*. Springer, New York. - Kuloğlu, B., & Özkan, E. (2021). On Generalized (
*k*,*r*)-Gauss Pell Numbers.*Journal of Science and Arts*, 3(56), 617–624. - Mikkawy, M., & Sogabe, T. (2010). A new family of
*k*-Fibonacci numbers.*Applied*

*Mathematics and Computation*, 215, 4456–4461. - Newman, M., Shanks, D., & Williams, H. C. (1980). Simple groups of square order and an interesting sequence of primes.
*Acta Arithmetica*, 38(2), 129–140. - Özkan, E. (2003). 3-Step Fibonacci Sequences in Nilpotent Groups.
*Applied Mathematics and Computation*, 144, 517–527. - Özkan, E., & Altun, I. (2019). Generalized Lucas Polynomials and Relationships Between the Fibonacci Polynomials and Lucas Polynomials.
*Communications in Algebra*, 47(10), 4020–4030. - Özkan, E., Altun, I., & Göçer, A. (2017). On Relationship among A New Family of
*k*-Fibonacci,*k*-Lucas Numbers, Fibonacci and Lucas Numbers.*Chiang Mai Journal of Science*, 44, 1744–1750. - Özkan, E., & Taştan, M. (2021). On a new family of Gauss
*k*-Lucas numbers and their polynomials.*Asian-European Journal of Mathematics*, 14(6), 2150101. - Panda, G. K., & Sahukar, M. K. (2020). Repdigits in Euler functions of associated Pell numbers.
*Proceedings – Mathematical Sciences*, 130, Article ID 25, 1–8. - Panwar, A., Sisodiya, K., & Rathore, G. P. S. (2017). On Some Identities for Generalized (
*k*,*r*)-Fibonacci Numbers.*International Journal of Mathematics Trends and Technology*, 51(2), 146–148. - Schuster, S., Fitchner, M., & Sasso, S. (2017). Use of Fibonacci numbers in lipidomics – Enumerating various classes of fatty acids.
*Scientific Reports*, 7, Article ID 39821. - Sloane, N. J. A. (2020). The On-line Encyclopedia of Integer Sequences, http://oeis.org/.
- Włoch, I., Bednarz, U., Brod, D., Włoch, A., & Wołowiec-Musial, M. (2013). On a new type of distance Fibonacci numbers.
*Discrete Applied Mathematics*, 161(16–17), 2695–2701.

### Manuscript history

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

## Related papers

## Cite this paper

Kuloğlu, B., & Özkan, E. (2022). On generalized (*k*, *r*)-Pell and (*k*, *r*)-Pell–Lucas numbers. *Notes on Number Theory and Discrete Mathematics*, 28(4), 765-777, DOI: 10.7546/nntdm.2022.28.4.765-777.