The double Fibonacci sequences in groups and rings

Ömür Deveci
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 24, 2018, Number 2, Pages 32–39
DOI: 10.7546/nntdm.2018.24.2.32-39
Full paper (PDF, 136 Kb)

Details

Authors and affiliations

Ömür Deveci
Department of Mathematics, Faculty of Science and Letters
Kafkas University 36100, Turkey

Abstract

In this paper, we define the double Fibonacci sequence and the basic double Fibonacci sequence in groups and rings. Then we examine these sequences in finite 2-generator groups and rings. Also, we obtain the periods of the double Fibonacci sequences and the basic double Fibonacci sequences in the dihedral group D_2m and the ring E for the generating pairs (a, b) and (b, a) as applications of the results obtained.

Keywords

• Double Fibonacci sequence
• Group
• Ring
• Period

2010 Mathematics Subject Classification

• 11B39
• 16P10
• 20D60

References

1. Aydın, H. & Dikici, R. (1998) General Fibonacci Sequences in Finite Groups, Fibonacci Quart., 36, 216–221.
2. Buschman, R. G. (1963) Fibonacci Numbers, Chebyshev Polynomials, Generalizations and Difference Equations, Fibonacci Quart., 1(4), 1–7.
3. Campbell, C. M., Campbell, P. P., Doostie, H. & Robertson, E. F. (2004) Fibonacci Lengths for Certain Metacyclic Groups, Algebra Colloq., 11, 215–222.
4. Campbell, C. M., Doostie, H. & Robertson, E. F. (1990) Fibonacci Length of Generating Pairs in Groups, Applications of Fibonacci Numbers, Springer Netherlands, 27–35.
5. DeCarli, D. J. (1970) A Generalized Fibonacci Sequence Over An Arbitrary Ring, Fibonacci Quart., 8(2), 182–184.
6. Deveci, O. & Karaduman, E. (2015) The Pell Sequences in Finite Groups, Util. Math., 96, 263–276.
7. Fine, B. (1993) Classification of Finite Rings of Order p2, Math. Mag., 66(4), 248–252.
8. Gopalan, M. A. & Geetha, V. (2014) Observations on Half-Companion Pell-Numbers, Int. J. Sci. Res. Publ., 4(2), 1–6.
9. Horadam, A. F. (1961) A Generalized Fibonacci Sequence, Amer. Math. Monthly, 68(5), 445–459.
10. Karaduman, E. & Aydın, H. (2004) On The Relationship Between The Recurrences in Nilpotent Groups and The Binomial Formula, Indian J. Pure Appl. Math., 35, 1093–1103.
11. Ozkan, E. (2014) Truncated Lucas Sequence and Its Period, Appl. Math. Comput., 232, 285–291.
12. Tasyurdu, Y. & Deveci, O. (2017) The Fibonacci Polynomials in Rings, Ars Combinatoria, 133, 355–366.
13. Vorobyov, N. N. (1963) The Fibonacci Numbers, Translated from Russian by Norman D. Whaland, J r., and Olga A. Tittlebaum, D. C. Heath and Co., Boston.