Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 26, 2020, Number 2, Pages 205—212
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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.
- Fibonacci number
- Hessenberg matrix
- Generalized modified Pell numbers
- Super-diagonal matrix
2010 Mathematics Subject Classification
- Daşdemir, A. (2011). On the Pell, Pell–Lucas and Modified Pell Numbers By Matrix Method, Appl. Math. Sci. (Ruse), 5 (64), 3173–3181.
- 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.
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- Koshy, T. (2001). Fibonacci and Lucas Numbers with Applications, Wiley, New York.
- Vajda, S. (1989). Fibonacci and Lucas Numbers, and the Golden Section, Theory and Applications, Ellis Horwood Ltd., Chichester.
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.