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Abstract
In this paper, we introduces Narayana sequence in two parameters, namely, (k, t)-Narayana sequence, which is generalization of classical Narayana sequence and provide some identities and matrix expressions. Further, we find relations between (k, t)-Narayana numbers and determinants and permanents of some Hessenberg matrices. We study recurrence relations and the sum of the first n terms of this sequence. We obtain some properties from matrices. Additionally, we define (k, t)−Narayana sequence for negative subscripts and derive some relations.
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References
- Wilmott, C.M. (2015) From Fibonacci Numbers to the Mathematics of Cows and Quantum Circuitry. Journal of Physics: Conference Series, 574:1-4
- Omotehinwa, T.O., Ramon, S.O. (2013) Fibonacci Numbers and Golden Ratio in Mathematics and Science. International Journal of Computer and Information Technology, 2(4):630-638
- Flaut, C., Shpakirskyi, V. (2013) On Generalised Fibonacci Quaternions and FibonacciNarayana Quaternions. Advances in Applied Clifford Algebras, 23(3):673-688
- Klein, S.T., Shapira, D. (2015) Random Access to Fibonacci Encoded Files. Discrete Applied Mathematics, 212(1):1-14
- Parajapat, S., Thakur, R.S. (2016) Realization of Information Exchange with Fibo-Q based Symmetric Cryptosystem. International Journal of Computer Science and Information Security, 14(2):216-222
- Barry, P., Hennessy, A. (2011) A Note on Narayana Triangles and Related Polynomials, Riordan Arrays, and MIMO Capacity Calculations. Journal of Integer Sequences, 14(1):1-26, Article 11.3.8
- Kirthi, K., Kak, S. (2016) The Narayana Universal Code. arXiv preprint arXiv:1601.07110.
- Das, M., Sinha, S. (2019) A Variant of the Narayana Coding Scheme. Control and Cybernetics, 48(3):473-484
- Ramirez, J.L., Sirvent, V.F. (2015) A Note on the k-Narayana Sequence. Annale Mathematicae et Informatice, 45(1):91-105
- Goy, T. (2018) On Identities with Multinomial Coefficients for Fibonacci-Narayana Sequence. Annale Mathematicae et Informatice, 49(1):75-84
- Ipek, A., Ari, K. (2014) On Hessenberg and Pentadiagonal Determinants Related with Fibonacci and Fibonacci-like Numbers. Applied Mathematics and Computation, 229(1):433-439
- Kilic, E., Stakhov, A.P. (2009) On the Fibonacci and Lucas p-Numbers, Their Sums, Families of Bipartite Graphs and Permanents of Certain Matrices. Choas, Solitons and Fractals, 40(5):2210-2221
- Mishra, V. (2022) Fibonacci sequence: History and Modern Applications. In: History and Development of Mathematics in India, Samikshika, National Mission for Manuscripts and DK Printworld (P) Ltd, New Delhi, 16:155-180
- Bala, R., Mishra, V. (2022) Narayana Matrix Sequence. Proceedings of the Jangjeon Mathematical Society, 25(4):427-434
- Bala, R., Mishra, V. (2022) Solutions of Equations x^2 − (p^2q^2 ± 3p)y^2 = ±k^t. Examples and Counterexamples, 2(1):1-4
- Bala, R., Mishra, V. (2021) On the Circulant Matrices with Ducci Sequence and Gaussian Fibonacci Numbers. In AIP Conference Proceedings, 2352(1):1-5, AIP Publishing LLC.
References
Wilmott, C.M. (2015) From Fibonacci Numbers to the Mathematics of Cows and Quantum Circuitry. Journal of Physics: Conference Series, 574:1-4
Omotehinwa, T.O., Ramon, S.O. (2013) Fibonacci Numbers and Golden Ratio in Mathematics and Science. International Journal of Computer and Information Technology, 2(4):630-638
Flaut, C., Shpakirskyi, V. (2013) On Generalised Fibonacci Quaternions and FibonacciNarayana Quaternions. Advances in Applied Clifford Algebras, 23(3):673-688
Klein, S.T., Shapira, D. (2015) Random Access to Fibonacci Encoded Files. Discrete Applied Mathematics, 212(1):1-14
Parajapat, S., Thakur, R.S. (2016) Realization of Information Exchange with Fibo-Q based Symmetric Cryptosystem. International Journal of Computer Science and Information Security, 14(2):216-222
Barry, P., Hennessy, A. (2011) A Note on Narayana Triangles and Related Polynomials, Riordan Arrays, and MIMO Capacity Calculations. Journal of Integer Sequences, 14(1):1-26, Article 11.3.8
Kirthi, K., Kak, S. (2016) The Narayana Universal Code. arXiv preprint arXiv:1601.07110.
Das, M., Sinha, S. (2019) A Variant of the Narayana Coding Scheme. Control and Cybernetics, 48(3):473-484
Ramirez, J.L., Sirvent, V.F. (2015) A Note on the k-Narayana Sequence. Annale Mathematicae et Informatice, 45(1):91-105
Goy, T. (2018) On Identities with Multinomial Coefficients for Fibonacci-Narayana Sequence. Annale Mathematicae et Informatice, 49(1):75-84
Ipek, A., Ari, K. (2014) On Hessenberg and Pentadiagonal Determinants Related with Fibonacci and Fibonacci-like Numbers. Applied Mathematics and Computation, 229(1):433-439
Kilic, E., Stakhov, A.P. (2009) On the Fibonacci and Lucas p-Numbers, Their Sums, Families of Bipartite Graphs and Permanents of Certain Matrices. Choas, Solitons and Fractals, 40(5):2210-2221
Mishra, V. (2022) Fibonacci sequence: History and Modern Applications. In: History and Development of Mathematics in India, Samikshika, National Mission for Manuscripts and DK Printworld (P) Ltd, New Delhi, 16:155-180
Bala, R., Mishra, V. (2022) Narayana Matrix Sequence. Proceedings of the Jangjeon Mathematical Society, 25(4):427-434
Bala, R., Mishra, V. (2022) Solutions of Equations x^2 − (p^2q^2 ± 3p)y^2 = ±k^t. Examples and Counterexamples, 2(1):1-4
Bala, R., Mishra, V. (2021) On the Circulant Matrices with Ducci Sequence and Gaussian Fibonacci Numbers. In AIP Conference Proceedings, 2352(1):1-5, AIP Publishing LLC.