Almost balancers, almost cobalancers, almost Lucas-balancers and almost Lucas-cobalancers

Ahmet Tekcan and Esra Zeynep Türkmen
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 29, 2023, Number 4, Pages 682–694
DOI: 10.7546/nntdm.2023.29.4.682-694
Full paper (PDF, 230 Kb)

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Authors and affiliations

Ahmet Tekcan
Bursa Uludag University, Faculty of Science
Department of Mathematics, Bursa, Türkiye

Esra Zeynep Türkmen
Bursa Uludag University, Faculty of Science
Department of Mathematics, Bursa, Türkiye

Abstract

In this work, the general terms of almost balancers, almost cobalancers, almost Lucas-balancers and almost Lucas-cobalancers of first and second type are determined in terms of balancing and Lucas-balancing numbers. Later some relations on all almost balancing numbers and all almost balancers are obtained. Further the general terms of all balancing numbers, Pell numbers and Pell–Lucas number are determined in terms of almost balancers, almost Lucas-balancers, almost cobalancers and almost Lucas-cobalancers of first and second type.

Keywords

• Balancing numbers
• Pell numbers
• Pell–Lucas numbers
• Almost balancing numbers

2020 Mathematics Subject Classification

• 11B37
• 11B39
• 11D09
• 11D79

References

1. Barbeau, E. J. (2003). Pell’s Equation. Springer-Verlag, New York.
2. Behera, A., Panda, G. K. (1999). On the square roots of triangular numbers. The Fibonacci Quarterly, 37(2), 98–105.
3. Flath, D. E. (1989). Introduction to Number Theory. Wiley.
4. Gozeri, G. K., Özkoç, A., & Tekcan, A. (2017). Some algebraic relations on balancing numbers. Utilitas Mathematica, 103, 217–236.
5. Kovacs, T., Liptai, K., & Olajos, P. (2010). On (a, b)-balancing numbers. Publicationes Mathematicae Debrecen, 77(3–4), 485–498.
6. Liptai, K. (2004). Fibonacci balancing numbers. The Fibonacci Quarterly, 42(4), 330–340.
7. Liptai, K. (2006). Lucas balancing numbers. Acta Mathematica et Informatica Universitatis Ostraviensis, 14, 43–47.
8. Liptai, K., Luca, F., Pinter, A., & Szalay, L. (2009). Generalized balancing numbers. Indagationes Mathematicae, 20(1), 87–100.
9. Olajos, P. (2010). Properties of balancing, cobalancing and generalized balancing numbers. Annales Mathematicae et Informaticae, 37, 125–138.
10. Panda, A. K. (2017). Some variants of the balancing sequences. Ph.D. thesis, National Institute of Technology Rourkela, India.
11. Panda, G. K., & Ray, P. K. (2011). Some links of balancing and cobalancing numbers with Pell and associated Pell numbers. Bulletin of the Institute of Mathematics, Academia Sinica, 6(1), 41–72.
12. Panda, G. K., & Ray, P. K. (2005). Cobalancing numbers and cobalancers. International Journal of Mathematics and Mathematical Sciences, 8, 1189–1200.
13. Panda, G. K., & Panda, A. K. (2015). Almost balancing numbers. Journal of the Indian Mathematical Society, 82(3–4), 147–156.
14. Ray, P. K. (2009). Balancing and cobalancing numbers. Ph.D. dissertation, Department of Mathematics, National Institute of Technology, Rourkela, India.
15. Szalay, L. (2007). On the resolution of simultaneous Pell equations. Annales Mathematicae et Informaticae, 34, 77–87.
16. Tengely, S. (2013). Balancing numbers which are products of consecutive integers. Publicationes Mathematicae Debrecen, 83(1–2), 197–205.
17. Özkoç, A., & Tekcan, A. (2017). On k-balancing numbers. Notes on Number Theory and Discrete Mathematics, 23(3), 38–52.
18. Tekcan, A. (2019). Sums and spectral norms of all almost balancing numbers. Creative Mathematics and Informatics, 28(2), 203–214.
19. Tekcan, A. (2019). Almost balancing, triangular and square triangular numbers. Notes on Number Theory and Discrete Mathematics, 25(1), 108–121.
20. Tekcan, A., & Erdem, A. (2020). t-cobalancing numbers and t-cobalancers. Notes on Number Theory and Discrete Mathematics, 26(1), 45–58.
21. Tekcan, A., & Aydın, S. (2021). On t-balancers, t-balancing numbers and Lucas t-balancing numbers. Libertas Mathematica, 41(1), 37–51.
22. Tekcan, A., & Erdem, A. (2023). General terms of all almost balancing numbers of first and second type. Communications in Mathematics, 31(1), 155–167.
23. Tekcan, A., & Yıldız, M. (2021). Balcobalancing numbers and balcobalancers. Creative Mathematics and Informatics, 30(2), 203–222.

Manuscript history

• Received: 24 November 2022
• Revised: 9 May 2023
• Accepted: 6 November 2023
• Online First: 13 November 2023

Copyright information

Ⓒ 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).