Relations on higher dimensional Padovan sequences

Renata Passos Machado Vieira, Francisco Regis Vieira Alves and Paula Maria Machado Cruz Catarino
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 27, 2021, Number 4, Pages 167–179
DOI: 10.7546/nntdm.2021.27.4.167-179
Full paper (PDF, 187 Kb)


Authors and affiliations

Renata Passos Machado Vieira
Department of Mathematics, Federal Institute of Education, Science and Technology of Ceara´
(IFCE) Fortaleza-CE, Brazil

Francisco Regis Vieira Alves
Department of Mathematics, Federal Institute of Education, Science and Technology of Ceara´
(IFCE) Fortaleza-CE, Brazil

Paula Maria Machado Cruz Catarino
Department of Mathematic, University of Tras-os-Montes and Alto Douro ´
Vila Real, Portugal


Many papers developed so far for Padovan sequences properties and its extensions usually follow the one-dimensional approach. The presented work introduces new relations for a higher dimensional sequence, this approach is adopted for two, three and n-dimensional Padovan Sequence. Several mathematical properties are discussed for the first time in the present work.


  • Relation
  • Dimensional
  • Padovan sequence
  • Recurrent relations
  • Two-dimensional.

2020 Mathematics Subject Classification

  • 11B37
  • 11B39


  1. Alfred, B. U. (1965). An Introduction to Fibonacci Discovery. The Fibonacci Association Cleveland, Cleveland, OH, U.S.A.
  2. Alves, F. R. V., & Catarino, P. M. M. C. (2017). A classe dos polinomios bivariados de Fibonacci (PBF): elementos recentes sobre a evolucao de um modelo. Revista Thema, 14(1), 112–136.
  3. Bilgici, G. (2017). Fibonacci and Lucas Sedenions. Journal of Integer Sequences, 20, 1–11.
  4. Catarino, P. M. M. C. (2018). Diagonal Functions of the k-Pell and k-Pell-Lucas Polynomials and some identities. Acta Mathematica Universitatis Comenianae, 87(1), 147–159.
  5. Chaves, A. P., Marques, D., & Togbé, A. (2012). On the sum of powers of terms of a linear recurrence sequence, Bulletin of the Brazilian Mathematical Society, 43(3), 397–406.
  6. Claudi, A., & Nelsen, R. B. (2015). A mathematical space odyssey: solid geometry in the 21th Century, American Mathematical Society; Edicao: UK ed., Washington.
  7. Harman, C. J. (1981). Complex fibonacci numbers, The Fibonacci Quarterly, 19, 82–86.
  8. Horadam, A. F. (1993). Quaternion recurrence relations. Ulam Quarterly, 2(2), 23–33.
  9. Jafari, M. (2016). On the Matrix Algebra of Complex Quaternions. Preprint. DOI: 10.13140/RG.2.1.3565.2321.
  10. Kantor, I. L., & Solodovnikov, A. S. (1989). Hypercomplex Numbers – An Elementary Introduction to Algebras Translated by A. Shenitzer, The Fibonacci Association Cleveland, Springer-Verlag, New York.
  11. Klavzar, S., & Mollard, M. (2012). Cube Polynomial of Fibonacci and Lucas Cubes. Acta Applicandae Mathematicae, 117, 93-105.
  12. Oliveira, R. R. de, Alves, F. R. V., & Paiva, R. E. B. (2017). Identidades bi e tridimensionais para os numeros de Fibonacci na forma complexa. C.Q.D.-Revista Eletronica Paulista de Matematica, 11, 91–106.
  13. Özdemir, G., Simsek, Y., & Milovanovic, G. V. (2017). Generating Functions for Special Polynomials and Numbers Including Apostol-Type and Humbert-Type Polynomials. Mediterranean Journal of Mathematics, 14(117), 1–16.
  14. Padovan, R. (2002). Dom Hans van der Laan and the plastic number. Nexus Network Journal, 4, 181–193.
  15. Polatlı, E., & Kesim, S. (2015). On quaternions with generalized Fibonacci and Lucas number components. Advances in Difference Equations, 169, 1–8.
  16. Savin, D. (2015). Some properties of Fibonacci numbers, Fibonacci octonions, and generalized Fibonacci–Lucas octonions, Advances in Difference Equations, 298, 1–10.
  17. Stewart, I. (1996). Tales of a Neglected Number. Mathematical Recreations – Scientific American, 274(6), 102–103.
  18. Voet, C., & Schoonjans, Y. (2012). Benedictine thought as a catalyst for 20th century liturgical space: the motivation behind Dom Hans van der Laan’s aesthetic church architecture, Proceeding of the 2nd international conference of the Europa Architectural History of Network, 255–261.
  19. Yılmaz, N., & Taskara, N. (2016). On the Properties of Iterated Binomial Transforms for the Padovan and Perrin Matrix Sequences. Mediterranean Journal of Mathematics, 13, 1435–1447.

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Cite this paper

Vieira, R. P. M., Alves, F. R. V., & Catarino, P. M. M. C. (2021). Relations on higher dimensional Padovan sequences. Notes on Number Theory and Discrete Mathematics, 27(4), 167-179, DOI: 10.7546/nntdm.2021.27.4.167-179.

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