On the connections between Fibonacci and Mulatu Numbers

Eudes Antonio Costa; Élis Gardel da Costa Mesquita; Paula M. M. C. Catarino

doi:10.22481/intermaths.v6i1.16742

Autores

Eudes Antonio Costa Universidade Federal do Tocantins, Arraias, Tocantins, BR https://orcid.org/0000-0001-6684-9961
Élis Gardel da Costa Mesquita Universidade Federal do Tocantins, Palmas, Tocantins, BR https://orcid.org/0000-0003-2385-4108
Paula M. M. C. Catarino Universidade de Trás-os-Montes e Alto Douro, Vila Real, Vila Real, PT https://orcid.org/0000-0001-6917-5093

DOI:

https://doi.org/10.22481/intermaths.v6i1.16742

Palavras-chave:

Generating function , Fibonacci numbers, Fibonacci-type numbers, Golden ratio

Resumo

In this work, we present a detailed study of the Fibonacci--Mulatu sequence, {FM_n}, defined recursively by FM_n+2=FM_n+1+FM_n with initial terms FM₀ = 4 and FM₁ = 1. We establish several properties of this sequence, elucidating the connections it evinces with the classical Fibonacci and Fibonacci--Lucas sequences. Additionally, we derive an expression for negative indexes. Specifically, we derive the Binet formula for this sequence and determine the limit for quotients involving both positive and negative indexes. Additionally, we construct three types of generating functions--ordinary, exponential, and Poisson--and examine classical identities, including those of d'Ocagne, Catalan, Cassini, Melham, Cesàro, as well as the convolution identity. We further calculate partial sums and alternating partial sums of this sequence.

Downloads

Não há dados estatísticos.

Referências

C. B. Boyer; U. C. Merzbach. História da matemática. São Paulo: Editora Blucher, 2019.

H. Eves. Introdução à história da matemática. São Paulo: Editora da UNICAMP, 2008.

H. E. Huntley. A divina proporção: um ensaio sobre a beleza na matemática. Brasília: Universidade de Brasília, 1985.

T. Koshy. Fibonacci and Lucas Numbers with Applications, vol. 1, John Wiley & Sons, 2019.

T. Roque. História da matemática. São Paulo: Editora Schwarcz-Companhia das Letras, 2012.

T. Roque e J. B. P. de Carvalho. Tópicos de história da matemática. Rio de Janeiro: Sociedade Brasileira de Matemática, 2012.

S. Vajda. Fibonacci and Lucas numbers, and the golden section: theory and applications. Courier Corporation, 2008.

N. N. Vorobiov. Números de Fibonacci: Lecciones Populares de Matemática. Moscou: Editora MIR, 1974.

G. A. P. Gusmão, “A sequência de Fibonacci.” Revista da Olimpíada - IME:UFG, vol. 4, pp. 55–71, 2003.

N. J. A. Sloane. The on-line encyclopedia of integer sequences. Sequence A002275. [S. l.]: The OEIS Foundation Inc., 2025. http://oeis.org/A094028

M. Lemma. “The fascinating and mysterious Mulatu numbers.” Advances and Applications in Mathematical Sciences, vol. 14, no. 2, 2015.

M. Lemma. “The Mulatu numbers.” Advances and Applications in Mathematical Sciences, vol. 10, no. 4, pp. 431–440, 2011.

M. Lemma. “Note on the three main theorems of the Mulatu numbers.” EPH-International Journal of Mathematics and Statistics, vol. 5, no. 2, pp. 12–15, 2019. https://doi.org/10.53555/eijms.v5i1.32

M. Lemma, Brown, K., Tanskely, L., et al. “A note on some interesting theorems of the Mulatu numbers.” EPH-International Journal of Mathematics and Statistics, vol. 5, no. 2, pp. 6–11, 2019.

M. Lemma and J. Lambright. “Our brief journey with some properties and patterns of the Mulatu numbers.” GPH-International Journal of Mathematics, vol. 4, no. 01, pp. 23–29, 2021.

M. Lemma, J. Lambright and A. Atena. “Some fascinating theorems of the Mulatu numbers.” Advances and Applications in Mathematical Sciences, vol. 15, no. 4, pp. 133–138, 2016.

M. Lemma., T. Muche, A. Atena., J. Lambright., and S. Dolo. “The Mulatu numbers, the newly celebrated numbers of pure mathematics.” GPH-International Journal of Mathematics, vol. 4, no. 11, pp. 12–18, 2021.

É. G. C. Mesquita, E. A. Costa, P. M. M. C. Catarino, F. R. Alves. “Jacobsthal-Mulatu Numbers." Latin American Journal of Mathematics, vol. 4 no. 1, pp. 23–45, 2025.

D. Patel and M. Lemma. “Using the gun of mathematical induction to conquer some theorems of the Mulatu numbers.” Journal of Advance Research in Mathematics and Statistics, vol. 4, no. 3, pp. 09–15, 2017.

A. Tiller and M. Lemma. “Using mathematical induction to conquer some theorems of the two famous celebrity numbers: the Fibonacci and Lucas.” Advances and Applications in Mathematical Sciences, vol. 15, no. 6, pp. 183–194, 2016.

A. Prabowo. “On generalization of Fibonacci, Lucas and Mulatu numbers.” International Journal of Quantitative Research and Modeling, vol. 1, no. 3, pp. 112–122, 2020.

F. Erduvan and M. G. Duman. “Mulatu numbers which are concatenation of two Fibonacci numbers.” Sakarya University Journal of Science, vol. 27, no. 5, pp. 1122–1127, 2023.

K. N. Adédji, R. J. Adjakidjè and A. Togbé. “Mulatu numbers as products of three generalized Lucas numbers.” Mathematica Pannonica, 2024.

R. P. Vieira, U. Parente, F. R. Alves, , and P. M. M. C. Catarino, “Uma nota da abordagem combinatória dos números de Mulatu.” Revista de Matemática da UFOP, vol. 1, no. 1, 2024.

A. T. Benjamin e J. J. Quinn. Proofs that really count: the art of combinatorial proof. Vol. 27, American Mathematical Society, 2022.

K. M. Nagaraja and P. Dhanya. “Identities on generalized Fibonacci and Lucas numbers.” Notes on Number Theory and Discrete Mathematics, vol. 3, no. 26, pp. 189–202, 2020.

Caminha, A. “Resolvendo recorrências.” Revista Matemática Universitária, vol. 1, pp. 23–31, 2019.

A. C. Morgado and P. C. P. Carvalho. Matemática discreta. Rio de Janeiro: SBM, 2015. Volume 16, Coleção Profmat.

K. H. Rosen. Discrete Mathematics and Its Applications. McGraw-Hill, 1999.

E. A. Costa, P. M. M. C. Catarino, F. S. Monteiro, V. M. A. Souza, and D. C. Santos, “Tricomplex Fibonacci numbers: a new family of Fibonacci-type sequences.” Mathematics, vol. 12, no. 23, p. 3723, 2024.