Nazmiye Yilmaz, Esra Kırmızı Çetinalp and Ömür Deveci
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 29, 2023, Number 2, Pages 226–240
DOI: 10.7546/nntdm.2023.29.2.226-240
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Authors and affiliations
Nazmiye Yilmaz
University of Karamanoğlu Mehmetbey, Kamil Özdağ Science Faculty,
Department of Mathematics, Turkey
Esra Kırmızı Çetinalp
University of Karamanoğlu Mehmetbey, Kamil Özdağ Science Faculty,
Department of Mathematics, Turkey
Ömür Deveci
Kafkas University, Faculty of Science and Letter, Department of Mathematics,
36100 Kars, Turkey
Abstract
In this paper, we define the six different quaternion-type cyclic-Fibonacci sequences and present some properties, such as, the Cassini formula and generating function. Then, we study quaternion-type cyclic-Fibonacci sequences modulo . Also we present the relationships between the lengths of periods of the quaternion-type cyclic-Fibonacci sequences of the first, second, third, fourth, fifth and sixth kinds modulo m and the generating matrices of these sequences. Finally, we introduce the quaternion-type cyclic-Fibonacci sequences in finite groups. We calculate the lengths of periods for these sequences of the generalized quaternion groups and obtain quaternion-type cyclic-Fibonacci orbits of the quaternion groups Q8 and Q16 as applications of the results.
Keywords
- Group
- Period
- Presentation
- Quaternion Fibonacci sequence
2020 Mathematics Subject Classification
- 11C20
- 11B39
- 11B50
- 20F05
- 20G20
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Manuscript history
- Received: 7 January 2023
- Revised: 4 April 2023
- Accepted: 24 April 2023
- Online First: 26 April 2023
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Ⓒ 2023 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).
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Cite this paper
Yilmaz, N., Çetinalp, E. K., & Deveci, Ö. (2023). The quaternion-type cyclic-Fibonacci sequences in groups. Notes on Number Theory and Discrete Mathematics, 29(2), 226-240, DOI: 10.7546/nntdm.2023.29.2.226-240.