The Generalization of Gaussians and Leonardo’s Octonions

Abstract

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.