Some Tridiagonal Matrix of the Repunit Sequence

Eudes Antonio Costa; Douglas Catulio dos Santos; Paula Maria Machado Cruz Catarino

doi:10.14244/lajm.v4i1.33

Autores

Eudes Antonio Costa Universidade Federal do Tocantins https://orcid.org/0000-0001-6684-9961
Douglas Catulio dos Santos Secretaria de Educação do Estado da Bahia https://orcid.org/0000-0002-5221-6087
Paula Maria Machado Cruz Catarino Universidade de Trás-os-montes e Alto Douro https://orcid.org/0000-0001-6917-5093

DOI:

https://doi.org/10.14244/lajm.v4i1.33

Palavras-chave:

Horadam sequence, Repunit sequence, Tridiagonal matrices

Resumo

This paper explores the connection between tridiagonal matrices and the repunit sequence, which is a type of Horadam sequence, and aims to establish new representations of repunit sequences using distinct tridiagonal matrices and their determinants. Motivated by some work that relates tridiagonal matrices to second-order linear recurrences, we present another representation of the repunit sequence by tridiagonal matrices.

Referências

[1] Sloane NJA, et al.. The on-line encyclopedia of integer sequences; 2003. Available from: https://oeis.org/.

[2] Santos DC, Costa EA. A note on Repunit number sequence. Intermaths. 2024;5(1):54-66. DOI: https://doi.org/10.22481/intermaths.v5i1.14922

[3] Horadam AF. A generalized Fibonacci sequence. The American Mathematical Monthly. 1961;68(5):455-9. DOI: https://doi.org/10.1080/00029890.1961.11989696

[4] Horadam AF. Basic properties of a certain generalized sequence of numbers. The Fibonacci Quarterly. 1965;3(3):161-76. DOI: https://doi.org/10.1080/00150517.1965.12431416

[5] Cerda G. Matrix methods in Horadam sequences. Boletíns de Matemáticas. 2012;19(2):97-106.

[6] Jaroma JH. Factoring Generalized Repunits. Bulletin of the Irish Mathematical Society. 2007;59:29-35. DOI: https://doi.org/10.33232/BIMS.0059.29.35

[7] Kalman D, Mena R. The Fibonacci numbers-exposed. Mathematics magazine. 2003;76(3):167-81. DOI: https://doi.org/10.1080/0025570X.2003.11953176

[8] Beiler AH. Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. 2nd ed. Cambridge MA: Dover; 1966.

[9] Costa EA, Santos DC. Algumas propriedades dos números monodígitos e das repunidades. Revista de Matemática da UFOP. 2022;2(1):47-58.

[10] Yates S. Repunits and repetends. Star Publishing Co., Inc; 1982.

[11] Costa EA, Santos DC, Monteiro FS, Souza VMA. On the repunit sequence at negative indices. Revista de Matemática da UFOP. 2024;1(1):1-12.

[12] Falcon S. On the generating matrices of the k-Fibonacci numbers. Proyecciones. 2013;32(4):347-57. DOI: https://doi.org/10.4067/S0716-09172013000400004

[13] Santos DC, Costa EA. Números repunidades e a representação matricial. Revista Sergipana de Matemática e Educação Matemática. 2024;9(1):81-96. DOI: https://doi.org/10.34179/revisem.v9i1.19491

[14] Cahill ND, D’enrico JR, Narayan DA, Narayan JY. Fibonacci Determinants. The College Mathematics Journal. 2002;33(3):221-5. DOI: https://doi.org/10.1080/07468342.2002.11921945

[15] Kilic E, Tasci D. On sums of second order linear recurrences by Hessenberg matrices. The Rocky Mountain Journal of Mathematics. 2008:531-44. DOI: https://doi.org/10.1216/RMJ-2008-38-2-531

[16] Kilic E, Tasci D. On the second order linear recurrences by tridiagonal matrices. Ars Combin. 2009;91:11-8.