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

Goksal Bilgici

doi:doi:10.11648/j.pamj.20130206.11

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

Goksal Bilgici

Published in Pure and Applied Mathematics Journal (Volume 2, Issue 6)

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.

Published in	Pure and Applied Mathematics Journal (Volume 2, Issue 6)
DOI	10.11648/j.pamj.20130206.11
Page(s)	174-178
Creative Commons	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.

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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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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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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://article.sciencepublishinggroup.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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TY  - JOUR
T1  - Generalized Order–k Pell–Padovan–Like Numbers by Matrix Methods
AU  - Goksal Bilgici
Y1  - 2013/11/30
PY  - 2013
N1  - https://doi.org/10.11648/j.pamj.20130206.11
DO  - 10.11648/j.pamj.20130206.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 174
EP  - 178
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20130206.11
AB  - 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.
VL  - 2
IS  - 6
ER  -

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Author Information

Goksal Bilgici

Department of Computer Education and Instructional Technology, Kastamonu University, Kastamonu, Turkey

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APA Style

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

Copy | Download

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

Copy | Download

@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://article.sciencepublishinggroup.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}
}

Copy | Download

TY  - JOUR
T1  - Generalized Order–k Pell–Padovan–Like Numbers by Matrix Methods
AU  - Goksal Bilgici
Y1  - 2013/11/30
PY  - 2013
N1  - https://doi.org/10.11648/j.pamj.20130206.11
DO  - 10.11648/j.pamj.20130206.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 174
EP  - 178
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20130206.11
AB  - 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.
VL  - 2
IS  - 6
ER  -

Copy | Download