<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>In order to explore the Leonardo sequence, the process of complex-ification 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 <italic>i</italic> 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.</p>
</abstract>

In order to explore the Leonardo sequence, the process of complex-ification 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.

The Generalization of Gaussians and Leonardo’s Octonions

Federal University of Ceará Fortaleza-CE

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

Annales Mathematicae Silesianae

This work is licensed under the Creative Commons Attribution 4.0 International License.

{"article-title":"The Generalization of Gaussians and Leonardo\u2019s Octonions"}

In order to explore the Leonardo sequence, the process of complex-ification of this sequence is carried out in this work. With this, the Gaussian and octonion...

The Generalization of Gaussians and Leonardo’s Octonions

Published Online: Feb 27, 2023

Page range: 117 - 137

Received: Jan 11, 2022

Accepted: Feb 06, 2023

DOI: https://doi.org/10.2478/amsil-2023-0004

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

© 2023 Renata Passos Machado Vieira et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

The Generalization of Gaussians and Leonardo’s Octonions

Published Online: Feb 27, 2023

Page range: 117 - 137

Received: Jan 11, 2022

Accepted: Feb 06, 2023

DOI: https://doi.org/10.2478/amsil-2023-0004

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

© 2023 Renata Passos Machado Vieira et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

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