On generalizations of the Jacobsthal sequence

Fügen Torunbalcı Aydın
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 24, 2018, Number 1, Pages 120–135
DOI: 10.7546/nntdm.2018.24.1.120-135
Full paper (PDF, 227 Kb)

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Authors and affiliations

Fügen Torunbalcı Aydın
Yildiz Technical University
Faculty of Chemical and Metallurgical Engineering
Department of Mathematical Engineering
Davutpasa Campus, 34220, Esenler, Istanbul, Turkey

Abstract

In this paper, the generalized Jacobsthal and generalized complex Jacobsthal and generalized dual Jacobsthal sequences using the Jacobsthal numbers are investigated. Also, special cases of these sequences are investigated. Furthermore, recurrence relations, vectors, the golden ratio and Binet’s formula for the generalized Jacobsthal sequences and generalized dual Jacobsthal sequences are given.

Keywords

• Jacobsthal number
• Jacobsthal–Lucas number
• Jacobsthal sequence
• Generalized Jacobsthal sequence
• Generalized complex Jacobsthal sequence
• Generalized dual Jacobsthal sequence

2010 Mathematics Subject Classification

• 11B37
• 11B50
• 11R52

References

1. Agrawal, O. M. P. (1987) Hamilton operators and dual-number-quaternions in spatial kinematics, Mechanism and Machine Theory, 22, 6, 569–575.
2. Atanassov, K. (2011) Remark on Jacobsthal numbers, Part 2. Notes on Number Theory and Discrete Mathematics, 17, 2, 37–39.
3. Atanassov, K. (2012) Short remarks on Jacobsthal numbers. Notes on Number Theory and Discrete Mathematics, 18, 2, 63–64.
4. Cerin, Z. (2007) Formulae for sums of Jacobsthal–Lucas numbers, Int. Math. Forum, 2, 1969–1984.
5. Cerin, Z. (2007) Sums of squares and products of Jacobsthal numbers, Journal of Integer Sequences, 10, Article 07.2.5.
6. Dasdemir, A. (2012) On the Jacobsthal numbers by matrix method, Fen Derg, 7, 1, 69–76.
7. Dasdemir, A. (2014) A study on the Jacobsthal and Jacobsthal–Lucas numbers by matrix method, DUFED Journal of Sciences, 3, 1, 13–18.
8. Djordjevic, G. B. (2000) Derivative sequences of generalized Jacobsthal and Jacobsthal– Lucas polynomials, Fibonacci Quarterly, 38, 4, 334–338.
9. Djordjevic, G. B. (2008) Generalized Jacobsthal polynomials, Fibonacci Quarterly, 38, 3, 239–243.
10. Gupta, V. K., & Panvar, Y. K. (2012) Common factors of generalized Fibonacci, Jacobsthal and Jacobsthal–Lucas numbers, International Journal of Applied Mathematical Research, 1, 4, 377–382.
11. Horadam, A. F. (1963) Complex Fibonacci numbers and Fibonacci quaternions, The American Mathematical Monthly, 70, 3, 289–291.
12. Horadam, A. F. (1988) Jacosthal and Pell curves, Fibonacci Quarterly, 26, 79–83.
13. Horadam, A. F. (1996) Jacobsthal representation numbers, Fibonacci Quarterly, 34, 40–54.
14. Horadam, A. F. (1997) Jacobsthal representation polynomials, Fibonacci Quarterly, 35, 137–148.
15. Koken, F., & Bozkurt, D. (2008) On the Jacobsthal numbers by matrix methods, Int. J. Contemp Math. Sciences, 3, 13, 605–614.
16. Koken, F., & Bozkurt, D. (2008) On the Jacobsthal–Lucas numbers by matrix methods, Int. J. Contemp Math. Sciences, 3, 13, 1629–1633.
17. Sloane, N. J. A. (1973) A Handbook of Integer Sequences, Academic Press.
18. Syznal-Liana, A., & Wloch, I. (2016) A note on Jacobsthal quaternions, Advances in Applied Clifford Algebras, 26, 1, 441–447.
19. Torunbalcı Aydın, F., & Yuce, S. (2017) A new approach to Jacobsthal quaternions, Filomat, 31, 18, 5567–5579.