**Hakan Akkuş, Ömür Deveci, Engin Özkan and Anthony G. Shannon**

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 30, 2024, Number 1, Pages 8–19

DOI: 10.7546/nntdm.2024.30.1.8-19

**Full paper (PDF, 944 Kb)**

## Details

### Authors and affiliations

Hakan Akkuş

*Department of Mathematics, Graduate School of Natural and Applied Sciences, Erzincan Binali Yıldırım University
Yalnızbağ Campus, 24100, Erzincan, Türkiye*

Ömür Deveci

*Department of Mathematics, Faculty of Sciences Arts, Kafkas University
36100 Kars, Türkiye*

Engin Özkan

*Department of Mathematics, Faculty of Sciences Arts, Erzincan Binali Yıldırım University
Yalnızbağ Campus, 24100, Erzincan, Türkiye*

Anthony G. Shannon

* Warrane College, The University of New South Wales
Kensington, NSW 2033, Australia*

### Abstract

Recursive sequences with gaps have been studied previously. This paper considers some elementary properties of such sequences where the gaps have been created on a regular basis from sequence to sequence – ‘discatenated’ (systematic gaps) and ‘lacunary’ (general gaps). In particular, their generating functions are developed in order to open up their general terms and relations with other properties.

### Keywords

- Fibonacci numbers
- Lucas sequences
- Lacunary
- Primordial sequences
- Recurrence relations

### 2020 Mathematics Subject Classification

- 11B39
- 11B0F

### References

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*k*-Fibonacci polynomials and generalized*k*-Lucas polynomials.*Notes on Number Theory and Discrete Mathematics*, 27(2), 148–158. - Whitney, R. E. (1970). On a class of difference equations.
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### Manuscript history

- Received: 8 January 2023
- Revised: 14 February 2023
- Accepted: 17 February 2024
- Online First: 23 February 2024

### Copyright information

Ⓒ 2024 by the Authors.

This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

## Related papers

- Craveiro, I. M., Spreafico, E. V. P., & Rachidi, M. (2023). New approaches of (
*q*,*k*)-Fibonacci–Pell sequences via linear difference equations. Applications.*Notes on Number Theory and Discrete Mathematics*, 29(4), 647–669. - Shannon, A. G. (2019). Applications of Mollie Horadam’s generalized integers to Fermatian and Fibonacci numbers.
*Notes on Number Theory and Discrete Mathematics*, 25(2), 113–126. - Taştan, M., Özkan, E., & Shannon, A. G. (2021). The generalized
*k*-Fibonacci polynomials and generalized*k*-Lucas polynomials.*Notes on Number Theory and Discrete Mathematics*, 27(2), 148–158.

## Cite this paper

Akkuş, H., Deveci, Ö, Özkan, E., & Shannon, A. G. (2024). Discatenated and lacunary recurrences. *Notes on Number Theory and Discrete Mathematics*, 30(1), 8-19, DOI: 10.7546/nntdm.2024.30.1.8-19.