The generalization of Gaussians and Leonardo's octonions

Renata Passos Machado Vieira; Milena Carolina dos Santos Mangueira; Francisco Régis Vieira Alves; Paula Maria Machado Cruz Catarino

Published: 2023-02-27

Vol. 37 No. 1 (2023)

The generalization of Gaussians and Leonardo's octonions

Renata Passos Machado Vieira

, Milena Carolina dos Santos Mangueira , Francisco Régis Vieira Alves , Paula Maria Machado Cruz Catarino

Abstract

In order to explore the Leonardo sequence, the process of complexification of this sequence is carried out in this work. With this, the Gaussian and octonion numbers of the Leonardo sequence are presented. Also, the recurrence, generating function, Binet’s formula, and matrix form of Leonardo’s Gaussian and octonion numbers are defined. The development of the Gaussian numbers is performed from the insertion of the imaginary component i in the one-dimensional recurrence of the sequence. Regarding the octonions, the terms of the Leonardo sequence are presented in eight dimensions. Furthermore, the generalizations and inherent properties of Leonardo’s Gaussians and octonions are presented.

Keywords:

matrix form , Binet’s formula , Gaussian , octonions , Leonardo sequence

Download files

PDF

Citation rules

Vieira, R. P. M., Mangueira, M. C. dos S., Alves, F. R. V., & Catarino, P. M. M. C. (2023). The generalization of Gaussians and Leonardo’s octonions. Annales Mathematicae Silesianae, 37(1), 117–137. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/15315

Similar Articles

Taras Goy, Mark Shattuck, Determinants of Toeplitz-Hessenberg matrices with generalized Leonardo number entries , Annales Mathematicae Silesianae: Vol. 38 No. 2 (2024)
Nur Şeyma Yilmaz, Andrzej Włoch, Engin Özkan, Dominik Strzałka, On doubled and quadrupled Fibonacci type sequences , Annales Mathematicae Silesianae: Vol. 38 No. 2 (2024)
Paula Catarino, Sandra Ricardo, On quaternion Gaussian Bronze Fibonacci numbers , Annales Mathematicae Silesianae: Vol. 36 No. 2 (2022)
Reza Farhadian, Rafael Jakimczuk, Notes on a general sequence , Annales Mathematicae Silesianae: Vol. 34 No. 2 (2020)
Reza Farhadian, Rafael Jakimczuk, A further generalization of $\lim_{n\to\infty}\sqrt[n]{n!}/n = 1/e$ , Annales Mathematicae Silesianae: Vol. 36 No. 2 (2022)
Reza Farhadian, Rafael Jakimczuk, A note on two fundamental recursive sequences , Annales Mathematicae Silesianae: Vol. 35 No. 2 (2021)
Göksal Bilgici, Dorota Bród, On r-Jacobsthal and r-Jacobsthal-Lucas numbers , Annales Mathematicae Silesianae: Vol. 37 No. 1 (2023)
Dorota Bród, Anetta Szynal-Liana, Iwona Włoch, One-parameter generalization of dual-hyperbolic Jacobsthal numbers , Annales Mathematicae Silesianae: Vol. 37 No. 2 (2023)
Dorota Bród, Anetta Szynal-Liana, Iwona Włoch, On a new generalization of Pell hybrid numbers , Annales Mathematicae Silesianae: Vol. 38 No. 2 (2024)
Dorota Bród, Adrian Michalski, On generalized Jacobsthal and Jacobsthal-Lucas numbers , Annales Mathematicae Silesianae: Vol. 36 No. 2 (2022)

1 2 3 4 5 6 7 8 9 > >>

You may also start an advanced similarity search for this article.

References

F.R.V. Alves and R.P.M. Vieira, The Newton fractal’s Leonardo sequence study with the Google Colab, Int. Elect. J. Math. Ed. 15 (2020), no. 2, Article No. em0575, 9 pp.
Google Scholar

F.R.V. Alves, R.P.M. Vieira, and P.M.M.C. Catarino, Visualizing the Newtons fractal from the recurring linear sequence with Google Colab: An example of Brazil X Portugal research, Int. Elect. J. Math. Ed. 15 (2020), no. 3, Article No. em0594, 19 pp.
Google Scholar

P. Catarino and A. Borges, On Leonardo numbers, Acta Math. Univ. Comenian. (N.S.) 89 (2020), no. 1, 75–86.
Google Scholar

C.J. Harman, Complex Fibonacci numbers, Fibonacci Quart. 19 (1981), no. 1, 82–86.
Google Scholar

A. Karataş and S. Halici, Horadam octonions, An. Ştiinţ. Univ. “Ovidius” Constanţa Ser. Mat. 25 (2017), no. 3, 97–106.
Google Scholar

O. Keçilioğlu and I. Akkus, The Fibonacci octonions, Adv. Appl. Clifford Algebr. 25 (2015), no. 1, 151–158.
Google Scholar

A.G. Shannon, A note on generalized Leonardo numbers, Notes Number Theory Discrete Math. 25 (2019), no. 3, 97–101.
Google Scholar

R.P.M. Vieira, F.R.V. Alves, and P.M.M.C. Catarino, Relações bidimensionais e identidades da sequência de Leonardo, Revista Sergipana de Matemática e Educação Matemática 4 (2019), no. 2, 156–173.
Google Scholar

R.P.M. Vieira, F.R.V. Alves, and P.M.M.C. Catarino, Uma extensão dos octônios de Padovan para inteiros não positivos, C.Q.D. – Revista Eletrônica Paulista de Matemática 19 (2020), Edição Dezembro, 9–16.
Google Scholar

R.P.M. Vieira, M.C. dos S. Mangueira, F.R.V. Alves, and P.M.M.C. Catarino, A forma matricial dos números de Leonardo, Ci. e Nat. 42 (2020), 40 yrs. – Anniv. Ed., Article No. e100, 6 pp.
Google Scholar

Renata Passos Machado Vieira

re.passosm@gmail.com
https://orcid.org/0000-0002-1966-7097

Affiliation:

Federal University of Ceará, Fortaleza-CE Brazil

Milena Carolina dos Santos Mangueira

Affiliation:

Federal Institute of Ceará, Fortaleza-CE Brazil

Francisco Régis Vieira Alves

Affiliation:

Federal Institute of Ceará, Fortaleza-CE Brazil

Paula Maria Machado Cruz Catarino

Affiliation:

University of Trás-os-Montes and Alto Douro, Vila Real Portugal

Vol. 37 No. 1 (2023)
Published: 2023-03-03

ISSN: 0860-2107

eISSN: 2391-4238

10.1515/amsil

Publisher

University of Silesia Press

Submit