On the Padovan p-circulant numbers

Güzel İpek, Ömür Deveci and Anthony G. Shannon
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 26, 2020, Number 3, Pages 224–233
DOI: 10.7546/nntdm.2020.26.3.224-233
Full paper (PDF, 233 Kb)

Details

Authors and affiliations

Güzel İpek
Department of Mathematics, Faculty of Science and Letters
Kafkas University, 36100 Kars, Turkey

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

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

Abstract

We construct three intercalated sequences and develop their essential properties which are generalizations of the three basic Fibonacci sequences. They are extensions of pulsated sequences described at previous Fibonacci conferences. We relate these sequences to the sequence {y_n}_{n ≥ 0} = {0, 1, 4, 15, 56, …}.

Keywords

• Padovan p-circulant sequence
• Matrix
• Representation

2010 Mathematics Subject Classification

• 11B50
• 11C20
• 20D60

References

1. Akuzum, Y., Deveci, Ö., & Shannon A. G. (2017). On the Pell p-circulant sequences, Notes Number Theory Disc. Math., 23 (2), 91–103.
2. Brualdi, R. A., & Gibson, P. M. (1997). Convex polyhedra of doubly stochastic matrices I: applications of permanent function, J. Combin. Theory, 22, 194–230.
3. Bueno, A. C. F. (2015). On right circulant matrices with general number sequence, Notes Number Theory Disc. Math., 21 (2), 55–58.
4. Chen, W. Y. C., & Louck, J. C. (1996). The combinatorial power of the companion matrix, Linear Algebra Appl., 232, 261–278.
5. Deveci, Ö. (2016). The Pell-Circulant sequences and their applications, Maejo Int. J. Sci. Technol., 10 (03), 284–293.
6. Deveci, Ö. (2018). The Padovan-circulant sequences and their applications, Math. Reports, 20 (70), 401–416.
7. Deveci, Ö., Akuzum, Y., & Karaduman, E. (2015). The Pell–Padovan p-sequences and its applications, Util. Math., 98, 327–347.
8. Deveci, Ö., & Karaduman, E. (2017). On the Padovan p-numbers, Hacettepe J. Math. Stat., 46 (4), 579–592.
9. Deveci, Ö., Karaduman, E., & Campbell, C. M. (2017). The Fibonacci-circulant sequences and their applications, Iranian J. Sci. Technol. Trans. A-Sci., 1–6.
10. Deveci, Ö., & Shannon, A. G. (2017). Pell–Padovan-circulant sequences and their applications, Notes Number Theory Disc. Math., 23 (3), 100–114.
11. Gogin, N. D., & Myllari, A. A. (2007). The Fibonacci–Padovan sequence and MacWilliams transform matrices, Program. Comput. Softw., published in Programmirovanie, 33 (2), 74–79.
12. Kalman, D. (1982). Generalized Fibonacci numbers by matrix methods, Fibonacci Quart., 20 (1), 73–76.
13. Kilic, E. (2009). The generalized Pell (p, i)-numbers and their Binet formulas, combinatorial representations, sums, Chaos, Solitons Fractals, 40 (4), 2047–2063.
14. Lancaster, P., & Tismenesky, M. (1985). The Theory of Matrices, Academic Press.
15. Lidl, R., & Niederreiter, H. (1986). Introduction to finite fields and their applications, Cambridge U. P.
16. Shannon, A. G. (1974). Some properties of a fundamental recursive sequence of arbitrary order, Fibonacci Quart., 12 (4), 327–335.
17. Shannon, A. G., & Horadam, A. F. (1971). Generating functions for powers of third order recurrence sequences, Duke Math. J., 38 (4), 791–794.
18. Stakhov, A. P., & Rozin, B. (2006). Theory of Binet formulas for Fibonacci and Lucas p-numbers, Chaos Solitons Fractals, 27, 1162–1167.
19. Tasci, D., & Firengiz, M. C. (2010). Incomplete Fibonacci and Lucas p-numbers, Math. Comput. Modelling., 52, 1763–1770.
20. Tuglu, N., Kocer, E. G., & Stakhov, A. P. (2011). Bivariate Fibonacci-like p-polynomials, Appl. Math. Comput., 217 (24), 10239–10246.