Combinatorial approach on the recurrence sequences: An evolutionary historical discussion about numerical sequences and the notion of the board

Francisco Regis Vieira Alves; Paula Maria Machado Cruz Catarino; Renata Passos Machado Vieira; Elen Viviani Pereira Spreafico

doi:10.29333/iejme/14387

Full Text (PDF)

Francisco Regis Vieira Alves ¹ ^* , Paula Maria Machado Cruz Catarino ² , Renata Passos Machado Vieira ³ , Elen Viviani Pereira Spreafico ⁴

More Detail

¹ Federal Institute of Education, Science and Technology of the State of Ceará–IFCE, Fortaleza, Ceará, BRAZIL² University of Trás-os-Montes and Alto Douro, Vila Real, PORTUGAL³ RENOEN/UFC, BRAZIL⁴ Federal University of Mato Grosso do Sul, Campo Grande, BRAZIL^* Corresponding Author

Abstract

The tradition of studies involving the combinatorial approach to recurring numerical sequences has accumulated a few decades of tradition, and several problems continue to attract the interest of mathematicians in several countries. This work specifically discusses the Fibonacci, Pell, and Jacobsthal sequences, focusing on Mersenne sequences. The often-used definition of board involves considering how to fill a specific regular surface -the board- with a limited quantity of regularly shaped tiles. On the other hand, an analogous problem can be generalized and exemplifies current research developments. Finally, the examples covered constitute unexpected ways of exploring visualization and other skills in mathematics teachers’ learning, consequently inspiring them for their teaching context.

Keywords

numerical sequences
combinatorial approach
board and tiles
mathematics teacher education

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

INT ELECT J MATH ED, Volume 19, Issue 2, May 2024, Article No: em0775

https://doi.org/10.29333/iejme/14387

Publication date: 01 Apr 2024

Online publication date: 22 Mar 2024

Article Views: 1147

Article Downloads: 986

Open Access References How to cite this article

References

Alves, F. R. V. (2017). Fórmula De Moivre, ou de Binet ou de Lamé: Demonstrações e generalidades sobre a sequência generalizada de Fibonacci–SGF [De Moivre, or Binet or Lamé Formula: Demonstrations and generalities about the generalized Fibonacci sequence–SGF]. Revista Brasileira de História da Matemática [Brazilian Journal of the History of Mathematics], 17(33), 1-16. https://doi.org/10.47976/RBHM2017v17n3301-16
Alves, F. R. V. (2022). Propriedades combinatórias sobre a sequência de Jacobsthal, a noção de tabuleiro e alguns apontamentos históricos [Combinatorial properties about the Jacobsthal sequence, the notion of a board and some historical notes]. Revista Cearense de Educação Matemática, 1(1), 1-13. https://doi.org/10.56938/rceem.v1i1.3146
Alves, F. R. V., & Catarino, P. M. C. (2022). A sequência de Padovan ou coordonier [The Padovan sequence or coordonier]. Revista de História da Matemática [History of Mathematics Magazine], 22(1), 85-97. https://doi.org/10.47976/RBHM2022v22n4521-43
Benjamin, A. T., & Quinn, J. J. (1999). Recounting Fibonacci and Lucas identities. The College Mathematics Journal, 30(5), 359-366. https://doi.org/10.2307/2687539
Benjamin, A. T., & Quinn, J. J. (2003a). Proofs that really count: The art of combinatorial proof. Mathematical Association of America. https://doi.org/10.5948/9781614442080
Benjamin, A. T., & Quinn, J. J. (2003b). The Fibonacci numbers: Exposed more discretely. Mathematics Magazine, 76(3), 182-192. https://doi.org/10.2307/3219319
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., Derks, H., & Quinn, J. J. (2011). The combinatorialization of linear recurrences. The Eletronic Journal of Combinatories, 18(2), 1-18. https://doi.org/10.37236/2008
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
Bicknell-Johnson, M. (1970). A primer for the Fibonacci numbers: Part VII. The Fibonacci Quarterly, 8(4), 407-420.
Bicknell-Johnson, M. (1987). A short history of the Fibonacci Quarterly. The Fibonacci Quarterly, 25(1), 2-6.
Bodeen, J., Butler, S., Kim, T., Sun, X., & Wang, S. (2014). Tiling a strip with triangles. The Eletronic Journal of Combinatorics, 21(1), 1-7. https://doi.org/10.37236/3478
Catarino, P. M. C., & Borges, A. (2020). On Leonardo numbers. Acta Mathematica Universitatis Comenianae [Mathematical Journal of the Comenian University], 89(1), 71-86.
Catarino, P., Campos, H., & Vasco, P. (2016). On the Mersenne sequence. Annales Mathematicae et Informaticae [Annals of Mathematics and Informatics], 46, 37-53.
Chelgham, M., & Boussayoud, A. (2021). On the k-Mersenne–Lucas numbers. Notes on Number Theory and Discrete Mathematics, 27(1), 7-13. https://doi.org/10.7546/nntdm.2021.27.1.7-13
Craveiro, I. M. (2004). Extensões e interpretações combinatórias para os números de Fibonacci, Pell e Jacobsthal [Extensions and combinatorial interpretations for Fibonacci, Pell and Jacobsthal numbers] [PhD thesis, Universidade Estadual de Campinas].
Diskaya, O., Menken, H., & Catarino, P. M. (2023). On the hyperbolic Leonardo and hyperbolic Francois Quaternions. New Journal of New Theroy, 42(3), 74-85. https://doi.org/10.53570/jnt.1199465
Došlić, T., & Podrug, L. (2022). Tilings of a honeycomb strip and higher order Fibonacci numbers. AirXiv, 25(2), 1-22. https://doi.org/10.48550/arXiv.2203.11761
Dresden, G., & Tulskikh, M. (2021). Tiling a (2×n)-board with dominos and L-shaped trominos. Journal of Integer Sequences, 24(6), 1-12.
Feinberg, M. (1963). Fibonacci-Tribonacci. The Fibonacci Quarterly, 1(3), 71-73.
Feng, J. (2011). More identities on the Tribonacci numbers. Ars Combinatorial, 2(3), 1-11.
Gould, H. W. (1981). A history of the Fibonacci Q-matrix and a higher-dimensional problem. The Fibonacci Quarterly, 19(3), 250-257.
Grimaldi, R. P. (2012). Fibonacci and Catalan numbers. Wiley & Sons. https://doi.org/10.1002/9781118159743
Gullberg, J. (1997). Mathematics: From the birth of numbers. W. W. Norton.
Koshy, T. (2001). Fibonacci and Lucas numbers with applications. Wiley. https://doi.org/10.1002/9781118033067
Koshy, T. (2014). Pell and Pell-Lucas numbers with applications. Springer. https://doi.org/10.1007/978-1-4614-8489-9
Koshy, T. (2019). Fibonacci and Lucas numbers with applications. Wiley & Sons. https://doi.org/10.1002/9781118742297
Lagrange, J. D. (2013). A combinatorial development of Fibonacci numbers in graph spectra. Linear Algebra and Applications, 438(11), 4335-4347. https://doi.org/10.1016/j.laa.2013.02.009
Sánchez, J. E. S. (2020). Sobre ladrilhamentos periódicos com polígonos regulares [About periodic tiling with regular polygons] [PhD thesis, Instituto de Matemática Pura e Aplicada–IMPA].
Soykan, Y. (2021). Binomial transform of the generalized third order Pell sequence. Communications in Mathematics and Applications, 12(1), 71-94. https://doi.org/10.26713/cma.v12i1.1371
Soykan, Y. (2022). Generalizes Erns numbers. Asian Journal of Pure and Applied Mathematics, 4(1), 136-150.
Spivey, M. Z. (2019). The art of proving binomial identities. Taylor & Francis Group. https://doi.org/10.1201/9781351215824
Spreacífico, E. V. P. (2014). Novas identidades envolvendo os números de Fibonacci, Lucas e Jacobsthal via ladrilhamentos [New identities involving Fibonacci, Lucas and Jacobsthal numbers via tiling] [PhD thesis, Universidade Estadual de Campinas].
Stillwell, J. (2010). Mathematics and its history. Springer. https://doi.org/10.1007/978-1-4419-6053-5
Tedford, S. J. (2019). Combinatorial identities for the Padovan numbers. The Fibonacci Quarterly, 57(4), 291-298.
Vieira, R. P. M., Mangueira, M. C. S., Sousa, R. C., Silva, J. G. A., & Alves, F. R. V. (2022). O processo de hibridização da sequência de Padovan: Uma experiência no curso de licenciatura em matemática no IFCE [The hybridization process of the Padovan sequence: An experience in the mathematics degree course at IFCE]. Revista Eletrônica de Educação Matemática [Electronic Magazine of Mathematics Education], 17(2), 1-22. https://doi.org/10.5007/1981-1322.2022.e88164
Vieira, R. P., & Alves, F. R. V. (2019). Sequences of Tridovan and their identities. Notes on Number Theory and Discrete Mathematics, 25(2), 185-197. https://doi.org/10.7546/nntdm.2019.25.3.185-197
Vorobiev, N. N. (2000). Fibonacci numbers. Springer.
Ziqian, J. (2019). Tetranacci identities with squares, dominoes, and hexagonal double-strips. AirXiv, 2(4), 1-20. https://doi.org/10.48550/arXiv.1907.09935
Ziqian, J., & Dresden, J. (2022). Tetranacci identities via hexagonal tilings. The Fibonacci Quarterly, 12(4), 1-15.

How to cite this article

APA

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), em0775. https://doi.org/10.29333/iejme/14387

Vancouver

Alves FRV, Catarino PMMC, Vieira RPM, Spreafico EVP. Combinatorial approach on the recurrence sequences: An evolutionary historical discussion about numerical sequences and the notion of the board. INT ELECT J MATH ED. 2024;19(2):em0775. https://doi.org/10.29333/iejme/14387

AMA

Alves FRV, Catarino PMMC, Vieira RPM, Spreafico EVP. Combinatorial approach on the recurrence sequences: An evolutionary historical discussion about numerical sequences and the notion of the board. INT ELECT J MATH ED. 2024;19(2), em0775. https://doi.org/10.29333/iejme/14387

Chicago

Alves, Francisco Regis Vieira, Paula Maria Machado Cruz Catarino, Renata Passos Machado Vieira, and Elen Viviani Pereira Spreafico. "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 2024 19 no. 2 (2024): em0775. https://doi.org/10.29333/iejme/14387

Harvard

Alves, F. R. V., Catarino, P. M. M. C., Vieira, R. P. M., and 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), em0775. https://doi.org/10.29333/iejme/14387

MLA

Alves, Francisco Regis Vieira et al. "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, vol. 19, no. 2, 2024, em0775. https://doi.org/10.29333/iejme/14387