Abstract
Keywords
Binet formula; Circulant matrix; Jacobsthal–Narayana sequence; Third-order sequence Vajda identity. Binet formula Circulant matrix Jacobsthal–Narayana sequence Third-order sequence Vajda identity
References
- [1] S. Vajda, Fibonacci and Lucas numbers, and the golden section: theory and applications, Courier Corporation, 2008
- [2] T. Koshy, Fibonacci and Lucas Numbers with Applications, Volume 2, John Wiley and Sons, 2019.
- [3] A. F. Horadam, Jacobsthal number representation, The Fibonacci Quarterly, Vol:34, No.1 (1996), 40-54.
- [4] Z. Cerin, Sums of squares and products of Jacobsthal numbers, Journal of Integer Sequences, Vol:10, (2007), 25.
- [5] K. T. Atanassov, (). Short remarks on Jacobsthal numbers, Notes on Number Theory and Discrete Mathematics, Vol:18, No.2 (2012), 63-64.
- [6] A. Das¸demir, On the Jacobsthal numbers by matrix method, Su¨leyman Demirel U¨ niversitesi Fen Edebiyat Faku¨ltesi Fen Dergisi, Vol:7, No.1 (2012), 69-76.
- [7] A. Das¸demir, A study on the Jacobsthal and Jacobsthal–Lucas numbers by matrix method, DUFED Journal of Sciences, Vol:3, No.1 (2014), 13-18.
- [8] S. H. J.Petroudi and M. Pirouz, On special circulant matrices with (k;h)-Jacobsthal sequence and (k;h)-Jacobsthal-like sequence, Int. J. Mathematics and scientific computation, Vol:6, No.1 (2016), 44-47.
- [9] T. Goy, On determinants and permanents of some Toeplitz-Hessenberg matrices whose entries are Jacobsthal numbers, Eurasian Mathematical Journal, Vol:9, No.4 (2018): p. 61-67.
- [10] A. Das¸demir, Mersene, Jacobsthal, and Jacobsthal-Lucas numbers with negative subscripts, Acta Mathematica Universitatis Comenianae, Vol:88, No.1 (2019), 142-156.
- [11] J. L. Ram´ırez and V. F. Sirvent, A note on the k-Narayana sequence, Ann. Math. Inform, Vol:45, (2015), 91-105.
- [12] G. Bilgici, The generalized order-k Narayana’s cows numbers, Mathematica Slovaca, Vol:66, No.4 (2016), 795-802.
- [13] Y. Soykan, On generalized Narayana numbers, Int. J. Adv. Appl. Math. Mech, Vol:7, No.3 (2020), 43-56.
- [14] F. Zhang, Matrix theory: basic results and techniques. New York: Springer, 2011.
Abstract
This paper introduces two new integer sequences that are the third-order recurrence relations. These are called Jacobsthal–Narayana and Jacobsthal-Lucas-Narayana sequences. In particular, great attention is focused on the identification of the Binet type representations for our new sequence, including the generating functions, some important identities, and generating matrix. Finally, we consider the circulant matrix whose entries are Jacobsthal–Narayana sequence and present an appropriate formula to find eigenvalues of that matrix.
Keywords
Binet formula Circulant matrix Jacobsthal–Narayana sequence Third-order sequence Vajda identity
References
- [1] S. Vajda, Fibonacci and Lucas numbers, and the golden section: theory and applications, Courier Corporation, 2008
- [2] T. Koshy, Fibonacci and Lucas Numbers with Applications, Volume 2, John Wiley and Sons, 2019.
- [3] A. F. Horadam, Jacobsthal number representation, The Fibonacci Quarterly, Vol:34, No.1 (1996), 40-54.
- [4] Z. Cerin, Sums of squares and products of Jacobsthal numbers, Journal of Integer Sequences, Vol:10, (2007), 25.
- [5] K. T. Atanassov, (). Short remarks on Jacobsthal numbers, Notes on Number Theory and Discrete Mathematics, Vol:18, No.2 (2012), 63-64.
- [6] A. Das¸demir, On the Jacobsthal numbers by matrix method, Su¨leyman Demirel U¨ niversitesi Fen Edebiyat Faku¨ltesi Fen Dergisi, Vol:7, No.1 (2012), 69-76.
- [7] A. Das¸demir, A study on the Jacobsthal and Jacobsthal–Lucas numbers by matrix method, DUFED Journal of Sciences, Vol:3, No.1 (2014), 13-18.
- [8] S. H. J.Petroudi and M. Pirouz, On special circulant matrices with (k;h)-Jacobsthal sequence and (k;h)-Jacobsthal-like sequence, Int. J. Mathematics and scientific computation, Vol:6, No.1 (2016), 44-47.
- [9] T. Goy, On determinants and permanents of some Toeplitz-Hessenberg matrices whose entries are Jacobsthal numbers, Eurasian Mathematical Journal, Vol:9, No.4 (2018): p. 61-67.
- [10] A. Das¸demir, Mersene, Jacobsthal, and Jacobsthal-Lucas numbers with negative subscripts, Acta Mathematica Universitatis Comenianae, Vol:88, No.1 (2019), 142-156.
- [11] J. L. Ram´ırez and V. F. Sirvent, A note on the k-Narayana sequence, Ann. Math. Inform, Vol:45, (2015), 91-105.
- [12] G. Bilgici, The generalized order-k Narayana’s cows numbers, Mathematica Slovaca, Vol:66, No.4 (2016), 795-802.
- [13] Y. Soykan, On generalized Narayana numbers, Int. J. Adv. Appl. Math. Mech, Vol:7, No.3 (2020), 43-56.
- [14] F. Zhang, Matrix theory: basic results and techniques. New York: Springer, 2011.
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Articles |
| Authors | |
| Early Pub Date | April 29, 2024 |
| Publication Date | April 30, 2024 |
| Submission Date | December 6, 2022 |
| Acceptance Date | October 23, 2023 |
| Published in Issue | Year 2024 Volume: 12 Issue: 1 |
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