Visualization and properties of boards and tiles: The case of the Padovan or Cordonier sequence

Francisco Alves; Renata Vieira; Paula Catarino; Milena Mangueira

doi:10.29333/mathsciteacher/16858

Full Text (PDF)

Francisco Alves ¹ ^* , Renata Vieira ² , Paula Catarino ³ , Milena Mangueira ¹

More Detail

¹ Federal Institute of Education, Science and Technology of the State of Ceará – IFCE, BRAZIL² Federal University of Ceará - UFC, BRAZIL³ University of Trás-os-Montes and Alto Douro – UTAD, PORTUGAL^* Corresponding Author

Abstract

The study of recurring numerical sequences usually involves a restricted context of circulation and communication of new ideas and abstract mathematical properties. On the other hand, authors of the history of mathematics books tend to emphasize, in an almost restrictive way, anecdotal curiosities about the Fibonacci sequence. Given this context, in this work, we find a proposal for exploring arithmetic and combinatorial properties using the notion of board and tiles, which correspond to properties of the Padovan or Cordonier numerical sequence. Through some elements, we will demonstrate that such an approach involves potential for application in the classroom to provide a relevant role for visualization and learning.

Keywords

number sequences
board and tiles
Padovan or Cordonier
mathematics teacher
visualization

License

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Research Article

Journal of Mathematics and Science Teacher, Volume 5, Issue 4, 2025, Article No: em086

https://doi.org/10.29333/mathsciteacher/16858

Publication date: 01 Oct 2025

Online publication date: 03 Sep 2025

Article Views: 454

Article Downloads: 150

Open Access References How to cite this article

References

Alves, F. R. V. (2017). A fórmula de de d´Moivre, ou de Binet ou de Lamé: Demonstrações e generalidades sore a sequência generalizada de Fibonacci [The formula of de Moivre, or of Binet or of Lamé: Demonstrations and generalities on the generalized Fibonacci sequence]. Revista Brasileira de História da Matemática, 17(33), 01-16. https://doi.org/10.47976/RBHM2017v17n3301-16
Alves, F. R. V. (2024). Números figurados gregos: O caso dos triangulares quadrados e sua recorrencia [Greek figurate numbers: The case of the triangular squares and their recurrence]. Revista de História da Matemática para Professores, 10(1), 1-12.
Alves, F. R. V., & Catarino, P. M. C. (2022). A sequência de Padovan ou Coordonier [The Padovan or Coordonier Sequence]. Revista Brasileira de História da Matemática, 22(45), 21-43. https://doi.org/10.47976/RBHM2022v22n4521-43
Alves, F. R. V., & Oliveira, P. C. C. (2024). Sobre os números p-ádicos: aspectos históricos, matemáticos e epistemológicos [On p-adic numbers: Historical, mathematical and epistemological aspects]. Revista Brasileira de História da Matemática, 24(48), 1-32. https://doi.org/10.47976/RBHM2024v24n481-32
Alves, F. R. V., Catarino, P. M. M. C., Vieira, R. P. M., & Spreafico, E. V. P. (2024). Combinatorial approach on the recurrence sequences: An evolutionary historical discussion about numerical sequences and the notion of the board. International Electronic Journal of Mathematics Education, 19(2), Article em0775. https://doi.org/10.29333/iejme/14387
Benjamin, A. T., & Quinn, J. J. (2003). Proofs that really count: The art of combinatorial proof. Mathematical Association of America.
Benjamin, A. T., & Walton, D. (2009). Counting on Chebyshev polynomials. Mathematics Magazine, 82(2), 117-126. https://doi.org/10.1080/0025570X.2009.11953605
Benjamin, A. T., Plott, S. S., & Sellers, J. A. (2008). Tiling proofs of recent sum identities involving Pell numbers. Annals of Combinatorics, 12(3), 271-278. https://doi.org/10.1007/s00026-008-0350-5
Bodeen, J., Butler, S., Kim, T., Sun, X., & Wang, S. (2014). Tiling a strip with triangles. The Eletronic Journal of Combinatorics, 21(1). https://doi.org/10.37236/3478
Catarino, P. M. C., & Borges, H. (2019). On Leonardo numbers. Acta Mathematica Universitatis Comenianae, 89(1), 75-86.
Cerda-Morales, G. (2025). On the Lichtenberg hybrid quaternions. Mathematica Moravica, 29(1), 31-41. https://doi.org/10.5937/MatMor2501031M
Chinn, P., Grimaldi, R., & Heubach, S. (2007). Tiling with L’s and Squares. Journal of Integer Sequences, 10(2), 1-17.
Costa, E. A., Catarino, P. M. M. C., Vasco, P., & Alves, F. R. V. (2025). A brief study on the k-dimensional repunit sequence. AXIOMS, 14(2), Article 109. https://doi.org/10.3390/axioms14020109
Craveiro, I. M. (2004). Extensões e Interpretações Combinatórias para os Números de Fibonacci, Pell e Jacobsthal [Combinatorial extensions and interpretations for the Fibonacci, Pell and Jacobsthal numbers](Doctoral dissertation, Campinas: Unicamp).
Došlić, T., & Podrug, L. (2022). Tilings of a honeycomb strip and higher order Fibonacci numbers. AirXiv, 25(2), 1-22.
Dresden, G., & Tulskikh, M. (2021). Tiling a (2 × n)-board with dominos and l-shaped trominos. Journal of Integer Sequence, 24(6), 1-12.
Dresden, J., & Ziqian, J. (2022). Tetranacci identities via hexagonal tilings. Fibonacci Quart, 12(4), 1-15.
Eves, H. (1969). An introduction to the history of mathematics (3^rd ed.). Holt, Reinhardt and Wilson ltd.
Graham, R., Patashnik, O., & Knuth, D. E. (1994). Concrete mathematics: A foundation for computer sence. Addison Wesley Pubb.
Grimaldi, R. (2012). Fibonacci and Catalan numbers: An introduction. Wiley. https://doi.org/10.1002/9781118159743
Gullberg, J. (1997). Mathematics: From the birth of numbers. Norton.
Hopkins, B. (2025). Hands-on combinatorics: Building colorful trains to manifest Pascal’s triangle, Fibonacci numbers, and Much more. American Mathematical Association.
Kleiner, I. (2012). Excursions in the history of mathematics. Springer. https://doi.org/10.1007/978-0-8176-8268-2
Snyder, W. M. (1982). Factoring repunits. The American Mathematical Monthly, 89(7), 462-466. Taylor & Francis. https://doi.org/10.1080/00029890.1982.11995478
Soykan, Y. (2023a). Generalized Pandita numbers. International Journal of Mathematics, Statistics and Operations Research, 3(1), 107-123.
Soykan, Y. (2023b). Generalized Richard numbers. International Journal of Advances in Appllied Mathematics and Mechanics, 10(3), 38-51.
Soykan, Y. (2023c). Generalized Pierre numbers. Journal of Progressive Research in Mathematics, 20(1), 16-38.
Soykan, Y. (2023d). Generalized Olivier numbers. Asian Research Journal of Mathematics, 19(1), 1-22. https://doi.org/10.9734/arjom/2023/v19i1634
Spreafico, E. V. P. (2014). Novas identidades envolvendo os números de Fibonacci, Lucas eJacobsthal via ladrilhamentos [New identities involving the Fibonacci, Lucas and Jacobsthal numbers via tilings] (Doctoral disertation, Universidad Estatal de Campinas). Instituto de Matemática.
Stakov, A. (2009). The mathematics of harmony: From Euclid to contemporary mathematics and computer science (Vol. 22). Word Scientific Press. https://doi.org/10.1142/6635
Tedford, S. J. (2019). Combinatorial identities for the Padovan numbers. The Fibonacci Quarterlly, 57(4), 291-298. https://doi.org/10.1080/00150517.2019.12427628
Vieira, R. P. M. (2024). Investigação da complexificação, generalização e modelo combinatório dos números de Padovan e Perrin com a engenharia didática [Investigation of the complexification, generalization and combinatorial model of the Padovan and Perrin numbers with didactic engineering] [Doctoral dissertation, Universidad Federal de Ceará].
Vieira, R. P. M., Alves, F. R. V., & Catarino, P. M. M. C. (2022). A combinatorial approach to the numbers of Padovan generalized. AXIOMS, 11, 1/1-16.

How to cite this article

APA

Alves, F., Vieira, R., Catarino, P., & Mangueira, M. (2025). Visualization and properties of boards and tiles: The case of the Padovan or Cordonier sequence. Journal of Mathematics and Science Teacher, 5(4), em086. https://doi.org/10.29333/mathsciteacher/16858

Vancouver

Alves F, Vieira R, Catarino P, Mangueira M. Visualization and properties of boards and tiles: The case of the Padovan or Cordonier sequence. Journal of Mathematics and Science Teacher. 2025;5(4):em086. https://doi.org/10.29333/mathsciteacher/16858

AMA

Alves F, Vieira R, Catarino P, Mangueira M. Visualization and properties of boards and tiles: The case of the Padovan or Cordonier sequence. Journal of Mathematics and Science Teacher. 2025;5(4), em086. https://doi.org/10.29333/mathsciteacher/16858

Chicago

Alves, Francisco, Renata Vieira, Paula Catarino, and Milena Mangueira. "Visualization and properties of boards and tiles: The case of the Padovan or Cordonier sequence". Journal of Mathematics and Science Teacher 2025 5 no. 4 (2025): em086. https://doi.org/10.29333/mathsciteacher/16858

Harvard

Alves, F., Vieira, R., Catarino, P., and Mangueira, M. (2025). Visualization and properties of boards and tiles: The case of the Padovan or Cordonier sequence. Journal of Mathematics and Science Teacher, 5(4), em086. https://doi.org/10.29333/mathsciteacher/16858

MLA

Alves, Francisco et al. "Visualization and properties of boards and tiles: The case of the Padovan or Cordonier sequence". Journal of Mathematics and Science Teacher, vol. 5, no. 4, 2025, em086. https://doi.org/10.29333/mathsciteacher/16858