THE RECURRENCE SEQUENCES VIA POLYHEDRAL GROUPS

Abstract

In this paper, we define recurrence sequences by using the relation matrices of the finite polyhedral groups and then, we obtain some of their properties. Also, we obtain the cyclic groups and the semigroups which are produced by the generating matrices when read modulo a and we study the sequences defined modulo a. Then we derive the relationships between the orders of the cyclic groups obtained and the periods of the sequences defined working modulo a. Furthermore, we extend these sequences to groups and obtain the periods of the sequences extended in the finite polyhedral groups case

Keywords

• Matrix, polyhedral sequences, polyhedral groups, period

References

1. Aydın, H. and Smith, G. C., Finite p-quotients of some cyclically presented groups, J. Lond. Math. Soc. 49 (1994), 83-92.
2. Bozkurt, D. and Tam, T-Y., Determinants and inverses of circulant matrices with Jacobsthal and Jacobsthal-Lucas numbers, Appl. Math. Comput. (2012), 219(2), 544-551.
3. Campbell, C. M. and Campbell, P.P., The Fibonacci lengths of binary polyhedral groups and related groups, Congr. Numer. 194 (2009), 95-102.
4. Coxeter, H. S. M. and Moser, W. O. J., Generators and relations for discrete groups, 3rd edition, Springer, Berlin, 1972.
5. Deveci, O., On the Fibonacci-circulant p-sequences, Util. Math. in press. Deveci, O. and Akuzum,Y., The recurrence sequences via Hurwitz matrices, Sci. Ann. “Al. I. Cuza” Univ. Iasi, in press. Deveci, O. and Karaduman, E., The cyclic groups via the Pascal matrices and the generalized Pascal matrices, Linear Algebra Appl. 437 (2012), 2538-2545.
6. Doostie, H. and Campbell, C. M., Fibonacci length of automorphism groups involving tri- bonacci numbers, Vietnam J. Math. (2000), 28(1), 57-65.
7. Falcon, S. and Plaza, A., k-Fibonacci sequences modulo m, Chaos Solitons Fractals (2009), (1), 497-504.
8. Frey, D. D. and Sellers, J. A., Jacobsthal numbers and alternating sign matrices, J. Integer Seq. 3 (2000), Article 00.2.3.

9. Gogin, N. D. and Myllari, A. A., The Fibonacci-Padovan sequence and MacWilliams trans- form matrices, Program. Comput. Softw., published in Programmirovanie (2007), 33(2), 74
10. Johnson, D .L., Topics in the theory of group presentations, London Math. Soc. Lecture Notes, Cambridge University Press, 1980.
11. Kalman, D., Generalized Fibonacci numbers by matrix methods, Fibonacci Quart. (1982), (1), 73-76.
12. Karaduman, E. and Deveci, O., k-nacci sequences in …nite triangle groups, Disc. Dyn. Nat. Soc. (2009), 453750-5-453750-10.
13. Kilic, E., The generalized Pell (p,i)-numbers and their Binet formulas, combinatorial repre- sentations, sums, Chaos, Solitons Fractals (2009), 40(4), 2047-2063.
14. Knox, S. W., Fibonacci sequences in …nite groups, Fibonacci Quart. (1992), 30(2), 116-120.
15. Lu, K. and Wang, J., k-step Fibonacci sequence modulo m, Util. Math. 71 (2006), 169-178.
16. Ozkan, E., On truncated Fibonacci sequences, Indian J. Pure Appl. Math. (2007), 38(4), 251.
17. Spinadel, V. W., The metallic means family and forbidden symmetries, Int. Math. J. (2002), (3), 279-288.
18. Stakhov, A. P. and Rozin, B., Theory of Binet formulas for Fibonacci and Lucas p-numbers, Chaos Solitons Fractals (2006), 27(5), 1162-1177.
19. Tasci, D. and Firengiz, M .C., Incomplete Fibonacci and Lucas p-numbers, Math. Comput. Modelling 52 (2010), 1763-1770.
20. Tuglu, N., Kocer, E. G. and Stakhov, A. P., Bivariate Fibonacci like p-polynomials, Appl. Math. Comput. (2011), 217(24), 10239-10246.
21. Wall, D. D., Fibonacci series modulo m , Amer. Math. Monthly 67 (1960), 525-532.
22. Wilcox, H.J., Fibonacci sequences of period n in groups, Fibonacci Quart. (1986), 24(4), 361.
23. Current address : Ömür Deveci: Faculty of Science and Letters, Kafkas University 36100, Turkey
24. E-mail address : [e-mail] ORCID: http://orcid.org/0000-0001-5870-5298
25. Current address : Ye¸sim Aküzüm: Faculty of Science and Letters, Kafkas University 36100, Turkey
26. E-mail address : [e-mail] ORCID: http://orcid.org/0000-0001-7168-8429
27. Current address : Colin M. Campbell: School of Mathematics and Statistics, University of St Andrews, North Haugh, St Andrews, Fife, KY16 9SS, Scotland E-mail address : [e-mail]