Ahmet Daşdemir

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 26, 2020, Number 2, Pages 205—212

DOI: 10.7546/nntdm.2020.26.2.205-212

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## Details

### Authors and affiliations

Ahmet Daşdemir

*Department of Mathematics, Faculty of Arts and Sciences
Kastamonu University, Kastamonu 37150, Turkey
*

### Abstract

This study shows that the generalized order-*k* Pell–Lucas and Modified Pell numbers can be expressed in terms of the well-known Fibonacci numbers. Certain *n*-square Hessenberg matrices with permanents equal to the Fibonacci numbers are defined. These Hessenberg matrices are then extended to super-diagonal (0,1,2)-matrices. In particular, the permanents of the super-diagonal matrices are shown to equal the components of the generalized order-*k* Pell–Lucas and Modified Pell numbers, and also their sums. In addition, two computer algorithms regarding our results are composed.

### Keywords

- Fibonacci number
- Hessenberg matrix
- Generalized modified Pell numbers
- Super-diagonal matrix
- Permanent

### 2010 Mathematics Subject Classification

- 11B39
- 15A15

### References

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- Kaygısız, K., & Şahin, A. (2012). Determinant and Permanent of Hessenberg Matrix and Fibonacci Type Numbers, Gen. Math. Notes., 9 (2), 32–41.
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- Vajda, S. (1989). Fibonacci and Lucas Numbers, and the Golden Section, Theory and Applications, Ellis Horwood Ltd., Chichester.

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## Cite this paper

Daşdemir, A. (2020). On generalized order-*k* modified Pell and Pell–Lucas numbers in terms of Fibonacci and Lucas numbers. Notes on Number Theory and Discrete Mathematics, 26 (2), 205-212, doi: 10.7546/nntdm.2020.26.2.205-212.