Generalized Pisano numbers

Yüksel Soykan, İnci Okumuş and Erkan Taşdemir
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 28, 2022, Number 3, Pages 477–490
DOI: 10.7546/nntdm.2022.28.3.477-490
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Authors and affiliations

Yüksel Soykan
Department of Mathematics, Faculty of Arts and Sciences, Zonguldak Bülent Ecevit University
67100, Zonguldak, Turkey

İnci Okumuş
Department of Engineering Sciences, Faculty of Engineering, Istanbul University-Cerrahpasa
34320 Istanbul, Turkey

Erkan Taşdemir
Pınarhisar Vocational School, Kırklareli University
39300, Kırklareli, Turkey

Abstract

In this paper, we define and investigate the generalized Pisano sequences and we deal with, in detail, two special cases, namely, Pisano and Pisano–Lucas sequences. We present Binet’s formulas, generating functions and Simson’s formulas for these sequences. Moreover, we give some identities and matrices associated with these sequences. Furthermore, we show that there are close relations between Pisano and Pisano–Lucas numbers and modified Oresme, Oresme–Lucas and Oresme numbers.

Keywords

  • Pisano numbers
  • Pisano–Lucas numbers
  • Tribonacci numbers
  • Modified Oresme numbers
  • Oresme–Lucas numbers
  • Oresme numbers

2020 Mathematics Subject Classification

  • 11B37
  • 11B39
  • 11B83

References

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  2. Clagett, M. (1981). “Oresme, Nicole”, C. C. Gillespie (ed.). Dictionary of Scientific
    Biography, Vol. 9, Charles Scribner’s Sons, New York, 223–230.
  3. Cook, C. K. (2004). Some sums related to sums of Oresme numbers, In: Howard F. T. (eds) Applications of Fibonacci Numbers. Proceedings of the Tenth International Research Conference on Fibonacci Numbers and their Applications, Kluwer Academic Publishers, Vol. 9, 87–99.
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  7. Soykan Y. (2020). A Study On Generalized (r, s, t)-Numbers. MathLAB Journal, 7, 101–129.
  8. Soykan, Y. (2021). A Study On Generalized p-Oresme Numbers. Asian Journal of Advanced Research and Reports, 15(7), 1–25.
  9. Soykan, Y. (2021). Generalized Oresme Numbers. Earthline Journal of Mathematical Sciences, 7(2), 333–367.
  10. Soykan, Y. (2021). On the Recurrence Properties of Generalized Tribonacci Sequence. Earthline Journal of Mathematical Sciences, 6(2), 253–269.

Manuscript history

  • Received: 26 February 2022
  • Revised: 23 July 2022
  • Accepted: 2 August 2022
  • Online First: 3 August 2022

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Cite this paper

Soykan, Y., Okumuş, İ., & Taşdemir, E. (2022). Generalized Pisano numbers. Notes on Number Theory and Discrete Mathematics, 28(3), 477-490, DOI: 10.7546/nntdm.2022.28.3.477-490.

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