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Generalized Order–k Pell–Padovan–Like Numbers by Matrix Methods

Received: 25 September 2013    Accepted:     Published: 30 November 2013
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Abstract

Weconsider the third – order recurrence relation Q_n=2Q_(n-2)+Q_(n-3) with initial conditionsQ_0=1,Q_1=0 "and" Q_2=2 and define these numbers as Pell – Padovan – like numbers.We extend this definition generalized order – k Pell – Padovan – like numbers and give some relations between thesenumbers and the Fibonacci numbers. Wealso obtain some relations of thesenumbers and matrices by using matrix methods.

DOI 10.11648/j.pamj.20130206.11
Published in Pure and Applied Mathematics Journal (Volume 2, Issue 6, December 2013)
Page(s) 174-178
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Fibonacci Sequence, Pell – Padovan’s Sequence, Generating Function, Binet Formula, Matrix Methods

References
[1] B.A. Brousseau, "Fibonacci Numbers and Geometry", Fibonacci Quart., vol. 10, no.3, pp. 303-318, 1972.
[2] M.C. Er, "Sums of Fibonacci Numbers by Matrix Method", Fibonacci Quart., vol.22, no.3, pp. 204-207 1984.
[3] D. Kalman, "Generalized Fibonacci Numbers by Matrix Method", Fibonacci Quart., vol. 20, no.1,pp. 73-76, 1982.
[4] K. Kaygisiz and D. Bozkurt, "k-Generalized Order-k Perrin Number Presentation by Matrix Method", ArsCombinatoria, vol.105, pp. 95-101, 2012.
[5] A.G. Shannon, A.F. Horadam and P. G. Anderson, "The Auxiliary Equation Associated with the Plastic Number", Notes on Number Theory and Discrete Mathematics, vol.12, no.1, pp. 1-12, 2006.
[6] A.G. Shannon, P G. Anderson and A.F. Horadam, "Properties of Cordonnier, Perrin and Van der Laan Numbers", International Journal of Mathematical Education in Science & Technology, vol. 37, no.7, pp. 825-831, 2006.
[7] F. Yilmaz, D. Bozkurt, "Some Properties of Padovan Sequence by Matrix Method", ArsCombinatoria, vol. 104, pp. 149-160, 2012.
[8] http://oeis.org, The Online Encyclopedia of Integer Sequences, Series: A008346.
Author Information
  • Department of Computer Education and Instructional Technology, Kastamonu University, Kastamonu, Turkey

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    Goksal Bilgici. (2013). Generalized Order–k Pell–Padovan–Like Numbers by Matrix Methods. Pure and Applied Mathematics Journal, 2(6), 174-178. https://doi.org/10.11648/j.pamj.20130206.11

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    ACS Style

    Goksal Bilgici. Generalized Order–k Pell–Padovan–Like Numbers by Matrix Methods. Pure Appl. Math. J. 2013, 2(6), 174-178. doi: 10.11648/j.pamj.20130206.11

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    AMA Style

    Goksal Bilgici. Generalized Order–k Pell–Padovan–Like Numbers by Matrix Methods. Pure Appl Math J. 2013;2(6):174-178. doi: 10.11648/j.pamj.20130206.11

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  • @article{10.11648/j.pamj.20130206.11,
      author = {Goksal Bilgici},
      title = {Generalized Order–k Pell–Padovan–Like Numbers by Matrix Methods},
      journal = {Pure and Applied Mathematics Journal},
      volume = {2},
      number = {6},
      pages = {174-178},
      doi = {10.11648/j.pamj.20130206.11},
      url = {https://doi.org/10.11648/j.pamj.20130206.11},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.pamj.20130206.11},
      abstract = {Weconsider the third – order recurrence relation Q_n=2Q_(n-2)+Q_(n-3) with initial conditionsQ_0=1,Q_1=0 "and" Q_2=2 and define these numbers as Pell – Padovan – like numbers.We extend this definition generalized order – k Pell – Padovan – like numbers and give some relations between thesenumbers and the Fibonacci numbers. Wealso obtain some relations of thesenumbers and matrices by using matrix methods.},
     year = {2013}
    }
    

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