The arrowhead-Pell-random-type sequences

Özgür Erdağ, Anthony G. Shannon and Ömür Deveci
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 24, 2018, Number 1, Pages 109–119
DOI: 10.7546/nntdm.2018.24.1.109-119
Full paper (PDF, 210 Kb)

Details

Authors and affiliations

Özgür Erdağ
Department of Mathematics, Faculty of Science and Letters
Kafkas University 36100, Turkey

Anthony G. Shannon
Fellow, Warrane College, The University of New South Wales
Kensington, 2033, Australia

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

Abstract

In this paper, we define the arrowhead-Pell-random-type sequence and then we obtain the generating function and the generating matrix of the sequence. Also, we derive the permanental, determinantal, combinatorial and exponential representations and the sums of the arrowhead-Pell-random-type numbers using the generating function and the generating matrix of the sequence.

Keywords

• The arrowhead-Pell numbers
• Sequence
• Matrix

2010 Mathematics Subject Classification

• 11B50
• 11C20
• 20D60

References

1. Akuzum, Y., Hiller, J., & Deveci, O. The Arrowhead-Pell Sequences, in press.
2. Atanassov, K. T., Deford, D. R., & Shannon, A. G. (2015) Pulsated Fibonacci recurrences. Fibonacci Quart., 52, 5, 22–27.
3. Brualdi, R. A., & Gibson, P. M. (1997) Convex polyhedra of doubly stochastic matrices I: Applications of permanent function, J. Combin. Theory, 22, 194–230.
4. Budden, M., Hiller, J., & Rapp, A. (2015) Generalized Ramsey theorems for r-uniform hypergraphs, Aust. J. Comb., 63, 1, 142–152.
5. Chen, W. Y. C., Louck, J.C. (1996) The combinatorial power of the companion matrix, Linear Algebra Appl., 232, 261–278.
6. Dawson, R., Gabor, G., Nowakowski, R., & Wiens, D. (1985) Random Fibonacci-type sequences, Fibonacci Quart., 23, 169–176.
7. Deveci, O. (2016) The Pell-circulant sequences and their applications, Maejo Int. J. Sci. Technol., 10(3), 284–293.
8. Deveci, O., & Shannon, A. G. (2017) On The Adjacency-Type Sequences, Int. J. Adv. Math., 2017, 2, 10–24.
9. Gultekin, I., & Deveci, O. (2016) On The Arrowhead-Fibonacci Numbers, Open Mathematics, 14, 1, 1104–1113.
10. Hofstadter, D. R. (1979) Godel, Escher, Bach: An eternal golden braid, Basic Books, NY.
11. Kalman, D. (1982) Generalized Fibonacci numbers by matrix methods, Fibonacci Quart., 20, 1, 73–76.
12. Kilic, E. (2009) The generalized Pell (p,i)-numbers and their Binet formulas, combinatorial representations, sums, Chaos, Solitons Fractals, 40, 4, 2047–2063.
13. Kilic, E., & Tasci, D. (2006) The Generalized Binet Formula, Representation and Sums of The Generalized Order-k Pell Numbers, Taiwan. J. Math., 10, 6, 1661–1670.
14. Lancaster, P., & Tismenetsky, M. (1985) The theory of matrices, Academic Press.
15. Lidl, R., & Niederreiter, H. (1994) Introduction to finite fields and their applications, Cambridge University Press.
16. Shannon, A. G., & Bernstein, L. (1973) The Jacobi-Perron Algorithm and the Algebra of Recursive Sequences, Bull. Australian Math. Soc., 8, 2, 261–277.
17. Shannon, A. G., & Horadam, A. F. (1994) Arrowhead curves in a tree of Pythagorean triples, Internat. J. Math. Ed. Sci.Technol., 25, 2, 255–261.
18. Sloane, N. J. A. Sequences A000045/M0692, A000073/M1074, A000078/M1108, A001591, A001622, A046698, A058265, A086088, and A118745, The On-Line Encyclopedia of Integer Sequences.
19. Stakhov, A. P., & Rozin, B. (2006) Theory of Binet formulas for Fibonacci and Lucas p-numbers, Chaos Solitons Fractals, 27, 1162–1167.
20. Tasci, D., & Firengiz, M. C. (2010) Incomplete Fibonacci and Lucas p-numbers, Math. Comput. Modelling, 52, 1763–1770.
21. Tasyurdu, Y., & Deveci, O. (2017) The Fibonacci Polynomials in Rings, Ars Comb., 133, 355–366.
22. Tasyurdu, Y., & Gultekin, I. (2013) On period of the sequence of Fibonacci polynomials modulo m, Discrete Dyn. Nat. Soc., 731482-1-731482-3.
23. Tuglu, N., Kocer, E.G., & Stakhov, A. P. (2011) Bivariate Fibonacci like p-polinomials, Appl. Math. Comput., 217, 24, 10239–10246.