Anthony G. Shannon, Özgür Erdağ and Ömür Deveci

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 27, 2021, Number 1, Pages 148–160

DOI: 10.7546/nntdm.2021.27.1.148-160

**Full paper (PDF, 235 Kb)**

## Details

### Authors and affiliations

Anthony G. Shannon

*Warrane College, University of New South Wales
Kensington, Australia
*

Özgür Erdağ

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

Ömür Deveci

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

### Abstract

In this paper, we define the Fibonacci–Pell *p*-sequence and then we discuss the connection of the Fibonacci–Pell *p*-sequence with the Pell and Fibonacci *p*-sequences. Also, we provide a new Binet formula and a new combinatorial representation of the Fibonacci–Pell *p*-numbers by the aid of the *n*-th power of the generating matrix of the Fibonacci–Pell *p*-sequence. Furthermore, we derive relationships between the Fibonacci–Pell *p*-numbers and their permanent, determinant and sums of certain matrices.

### Keywords

- Pell sequence
- Fibonacci
*p*-sequence - Matrix
- Representation

### 2010 Mathematics Subject Classification

- 11K31
- 11C20
- 15A15

### References

- Bradie, B. (2010). Extension and refinements of some properties of sums involving Pell number. Missouri Journal of Mathematical Sciences, 22(1), 37–43.
- Brualdi, R. A., & Gibson, P. M. (1977). Convex polyhedra of doubly stochastic matrices. I. Applications of permanent function, Journal of Combinatorial Theory, Series A, 22(2), 194–230.
- Chen, W. Y. C., & Louck, J. D. (1996). The combinatorial power of the companion matrix. Linear Algebra and its Applications, 232, 261–278.
- Devaney, R. L. (1999). The Mandelbrot set, the Farey tree, and the Fibonacci sequence, The American Mathematical Monthly, 106(4), 289–302.
- Deveci, O., Adiguzel, Z. & Dogan, T. (2020). On the Generalized Fibonacci-circulant-Hurwitz Numbers.
*Notes on Number Theory and Discrete Mathematics*, 26(1), 179–190. - Deveci, O., & Artun, G. (2019). On the Adjacency-Jacobsthal Numbers. Communications in Algebra, 47 (11), 4520-4532.
- Deveci, O., Karaduman, E., & Campbell, C. M. (2017). The Fibonacci-Circulant Sequences and Their Applications. Iranian Journal of Science and Technology, Transaction A, Science, 41(4), 1033–1038.
- Frey, D. D., & Sellers, J. A. (2000). Jacobsthal numbers and alternating sign matrices. Journal of Integer Sequences, 3, Article 00.2.3.
- 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.
- Horadam, A. F. (1994). Applications of modified Pell numbers to representations. Ulam Quarterly Journal, 3(1), 34–53.
- Johnson, R. C. (2009). Fibonacci numbers and matrices, Available online: https://maths.dur.ac.uk/~dma0rcj/PED/fib.pdf.
- Kalman, D. (1982). Generalized Fibonacci numbers by matrix methods. The Fibonacci Quarterly, 20(1), 73–76.
- Kilic, E. (2008). The Binet fomula, sums and representations of generalized Fibonacci
*p*-numbers. European Journal of Combinatorics, 29(3), 701–711. - Kilic, E., & Tasci, D. (2006). The generalized Binet formula, representation and sums of the generalized order-
*k*Pell numbers. Taiwanese Journal of Mathematics, 10(6), 1661–1670. - Kocer, E. G., & Tuglu, N. (2007). The Binet formulas for the Pell and Pell–Lucas
*p*-numbers. Ars Combinatoria, 85, 3–17. - Koken, F., & Bozkurt, D. (2008). On the Jacobsthal numbers by matrix methods. International Journal of Contemporary Mathematical Sciences, 3(13), 605–614.
- Lancaster, P. & Tismenetsky, M. (1985). The Theory of Matrices: with Applications, Elsevier.
- Lidl, R., & Niederreiter, H. (1994). Introduction to Finite Fields and Their Applications, Cambridge University Press.
- McDaniel, W. L. (1996). Triangular numbers in the Pell sequence. The Fibonacci Quarterly, 34(2), 105–107.
- Shannon, A. G., Anderson, P. G., & Horadam, A. F. (2006). Properties of Cordonnier, Perrin and Van der Laan numbers. International Journal of Mathematical Education in Science and Technology, 37(7), 825–831.
- Shannon, A. G., Horadam, A. F., & Anderson, P. G. (2006). The auxiliary equation associated with the plastic number.
*Notes on Number Theory and Discrete Mathematics*, 12(1), 1–12. - Stakhov, A. P. (1999). A generalization of the Fibonacci Q-matrix. Rep. Natl. Acad. Sci. Ukraine, 9, 46–49.
- Stakhov, A. P., & Rozin, B. (2006). Theory of Binet formulas for Fibonacci and Lucas
*p*-numbers. Chaos, Solitions, Fractals, 27(5), 1162–1177. - Stewart, I. (1996). Tales of a neglected number. Scientific American, 274(6), 102–103.
- Tasci, D., & Firengiz, M. C. (2010). Incomplete Fibonacci and Lucas
*p*-numbers.

Mathematical and Computer Modelling, 52(9–10), 1763–1770..

## Related papers

- Shannon, A. G., Horadam, A. F., & Anderson, P. G. (2006). The auxiliary equation associated with the plastic number.
*Notes on Number Theory and Discrete Mathematics*, 12(1), 1–12. - Deveci, Ö., Adigüzel, Z. & Doğan, T. (2020). On the Generalized Fibonacci-circulant-Hurwitz numbers. Notes on Number Theory and Discrete Mathematics, 26(1), 179–190.
- Mehraban, E., & Hashemi, M. (2023). Coding theory on the generalized balancing sequence.
*Notes on Number Theory and Discrete Mathematics*, 29(3), 503-524.

## Cite this paper

Shannon, A.G., Erdağ, Ö., & Deveci, Ö. (2021). On the connections between Pell numbers and Fibonacci *p*-numbers. *Notes on Number Theory and Discrete Mathematics*, 27(1), 148-160, DOI: 10.7546/nntdm.2021.27.1.148-160.