Leonardo’s three-dimensional relations and some identities

Renata Passos Machado Vieira, Milena Carolina dos Santos Mangueira, Francisco Regis Vieira Alves and Paula Maria Machado Cruz Catarino
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 27, 2021, Number 4, Pages 32—42
DOI: 10.7546/nntdm.2021.27.4.32-42
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Authors and affiliations

Renata Passos Machado Vieira
Department of Mathematics, Federal Institute of Education, Science and Techonology
of State of Ceara – IFCE, Treze of Maio, Brazil

Milena Carolina dos Santos Mangueira
Department of Mathematics, Federal Institute of Education, Science and Techonology
of State of Ceara – IFCE, Treze of Maio, Brazil

Francisco Regis Vieira Alves
Department of Mathematics, Federal Institute of Education, Science and Techonology
of State of Ceara – IFCE, Treze of Maio, Brazil

Paula Maria Machado Cruz Catarino
University of Tras-os-Montes and Alto Douro – UTAD
Vila Real, Portugal

Abstract

In this work, new results are explored in relation to the Leonardo sequence. With that, a study about this second order recursive sequence, little explored in the mathematical scope, is briefly presented, relating it to the Fibonacci sequence. Thus, its complexification process is carried out, where from its one-dimensional model, imaginary units are inserted, obtaining Leonardo’s three-dimensional numbers. In this way, the imaginary units i and j are inserted. Finally, some three-dimensional identities are presented for Leonardo’s numbers.

Keywords

  • Two-dimensional relations
  • Three-dimensional relations
  • Leonardo sequence
  • Fibonacci sequence

2020 Mathematics Subject Classification

  • 11B37
  • 11B39

References

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  5. Oliveira, R. R., Alves, F. R. V., & Paiva, R. E. B. (2017). Identidades bi e tridimensionais para os numeros de Fibonacci na forma complexa. CQD-Revista Eletronica Paulista de Matematica, Bauru, 11, 91–106.
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  7. Vieira, R. P. M., Alves, F. R. V., & Catarino, P. M. M. C. (2019). Relacoes bidimensionais e identidades de Leonardo. Revista Sergipana de Matematica e Educacao Matematica, 4(2), 156–173.

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Cite this paper

Vieira, R. P. M., Mangueira, M. C. S., Alves, F. R. V., & Catarino, P. M. M. C. (2021). Leonardo’s three-dimensional relations and some identities. Notes on Number Theory and Discrete Mathematics, 27(4), 32-42, doi: 10.7546/nntdm.2021.27.4.32-42.

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