The Pell-Fibonacci Sequence Modulo m

Year 2020, Volume: 5 Issue: 3, 280 - 284, 30.12.2020

Yeşim Aküzüm

Abstract

In [6], Deveci defined the Pell-Fibonacci sequence as follows:
P − F (n + 4) = 3P − F (n + 3) − 3P − F (n + 1) − P − F (n)
for n ≥ 0 with initial constants P − F (0) = P − F (1) = P − F (2) = 0,P − F (3) = 1. Also, he derived the permanental and determinantal representations of the Pell-Fibonacci numbers and he obtained miscellaneous properties of the Pell-Fibonacci numbers by the aid of the generating function and the generating matrix of the Pell-Fibonacci sequence. The linear recurrence sequences appear in modern research in many fields from mathematics, physics, computer, architecture to nature and art; see, for example, [2, 4, 13, 18]. In this paper, we obtain the cyclic groups which are produced by generating matrix of the Pell-Fibonacci sequence when read modulo m. Furthermore, we research the Pell-Fibonacci sequence modulo m, and then we derive the relationship between the order the cyclic groups obtained and the periods of the Pell-Fibonacci sequence modulo m.

Keywords

The Pell-Fibonacci sequence, Modulo, Period

References

• References1 Akuzum Y, Deveci O, Shannon AG. On The Pell p-Circulant Sequences. Notes Number Theory Disc. Math. 23(2), 2017, 91-103.
• References2 Alexopoulos T, Leontsinis S. Benford’s Law in Astronomy. J. Astrophysics Astronomy. 35, 2014, 639-648.
• References3 Aydin H, Dikici R. General Fibonacci sequences in finite groups. Fibonacci Quart. 36(3), 1998, 216-221.
• References4 Bruhn H, Gellert L, Gunther J. Jacobsthal Numbers in Generalised Petersen Graphs. Electronic Notes Disc. Math. 9, 2015, 465-472.
• References5 Campbel CM, Doostie H, Robertson EF. Fibonacci Length of Generating Pairs in Groups, in Applications of Fibonacci Numbers. Vol. 3 Eds. G. E. Bergum et al. Kluwer Academic Publishers, 1990, 27-35.
• References6 Deveci O. On The Connections Between Fibonacci, Pell, Jacobsthal and Padovan Numbers, is submitted.
• References7 Deveci O. The Pell-Padovan Sequences and The Jacobsthal-Padovan Sequences in Finite Groups. Util. Math. 98, 2015, 257-270.
• References8 Deveci O. The Pell-Circulant Sequences and Their Applications. Maejo Int. J. Sci. Technol. 10, 2016, 284-293.
• References9 Deveci O, Akuzum Y. The Cyclic Groups and The Semigroups via MacWilliams and Chebyshev Matrices. Journal Math. Research. 6(2), 2014, 55-58.
• References10 Deveci O, Akuzum Y, Karaduman E. The Pell-Padovan p-sequences and its applications. Util. Math. 98, 2015, 327-34.
• References11 Deveci O, Akuzum Y, Karaduman E, Erdag E. The Cyclic Groups via Bezout Matrices. Journal Math. Research. 7(2), 2015, 34-41.
• References12 Doostie H, Campbell P. On the Commutator Lengths of Certain Classes of Finitely Presented Groups. Int. J. Math. Math. Sci. 2006, 1-9.
• References13 Iwaniec H. On The Problem of Jacobsthal. Demonstratio Math. 11, 1978, 225-231.
• References14 Karaduman E, Aydin H. On Fibonacci Sequences in Nilpotent Groups. Math. Balkanica, 17, 2003, 207-214.
• References15 Knox SW. Fibonacci sequences in finite groups. Fibonacci Quart.,30(2), 1992, 116-120.
• References16 Lu K, Wang J. ¨ k-step Fibonacci Sequence Modulo m. Util. Math. 71, 2007, 169-178.
• References17 Ozkan E, Aydin H, Dikici R. 3-step Fibonacci series modulo m. Appl. Math. Comput. 143, 2003, 165-172.
• References18 Pighizzini G, Shallit J. Unary Language Operations, State Complexity and Jacobsthal’s Function. Int. J. Foundations Comp. Sci. 13, 2002, 145-159.
• References19 Wall DD. Fibonacci series modulo m. Amer. Math. Monthly, 67, 1960, 525-532.
• References20 Wilcox H.J. Fibonacci sequences of period n in groups. Fibonacci Quart. 24, 198.