On generalized order-k modified Pell and Pell–Lucas numbers in terms of Fibonacci and Lucas numbers

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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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

  1. Daşdemir, A. (2011). On the Pell, Pell–Lucas and Modified Pell Numbers By Matrix Method, Appl. Math. Sci. (Ruse), 5 (64), 3173–3181.
  2. Daşdemir, A. (2016). Generalizations of Modified Pell and Pell–Lucas Sequences and Their Generating Matrices and Some Sums, Erzincan Univ J Sci Tech., 9 (3), 178–184.
  3. Ercolano, J. (1994). Matrix Generators of Pell Sequences, Fibonacci Quart., 3 (1), 34–53.
  4. Horadam, A. F. (1971). Pell identities, Fibonacci Quart., 9 (3), 245–263.
  5. Horadam, A. F. (1994). Applications of Modified Pell Numbers to Representations, Ulam Quart. 3 (1), 34–53.
  6. Kaygısız, K., & Şahin, A. (2012). Determinant and Permanent of Hessenberg Matrix and Fibonacci Type Numbers, Gen. Math. Notes., 9 (2), 32–41.
  7. Koshy, T. (2001). Fibonacci and Lucas Numbers with Applications, Wiley, New York.
  8. 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.

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