Perrin n-Dimensional Relations

Abstract

This work aims, to perform a complexity in the Perrin sequence, to present the two-dimensional, three-dimensional, and n-dimensional recurrence relations of this sequence. Thus, from the one-dimensional relationship of this sequence, we will discuss the increase of its dimensionality and the insertion of imaginary units in the Perrin sequence, which is a recursive sequence of third order and presents large similarities with the Padovan sequence, differing only its initial values. Moreover, we will present a relationship between the Perrin numbers and the Padovan numbers, which will be used to perform the complexity of this sequence.

Keywords

• n-dimensional relations
• Perrin sequence
• Three-dimensional relations
• Two-dimensional relations

References

1. [1] M. C. dos S. Mangueira, R. P. M. Vieira, F. R. V. Alves, P. M. M. C. Catarino, A generalizacao da forma matricial da sequencia de Perrin, ReviSeM, 5 (1) (2020), 384-392.
2. [2] M. C. dos S. Mangueira, R. P. M. Vieira, F. R. V. Alves, P. M. M. C. Catarino, A generalized Perrin polynomial sequence and its two-dimensional recurrences, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.), in press.
3. [3] P. Seenukul, Matrices which have similar properties to Padovan q -matrix and its generalized relations, SNRU Journal of Science and Technology, 7 (2) (2015), 90-94.
4. [4] A. G. Shannon, P. G. Anderson, A. F. Horadam, Properties of Cordonnier, Perrin and van der Laan numbers, IJEMST, 37 (7) (2006), 825-831.
5. [5] K. Sokhuma, Matrices formula for padovan and perrin sequences, Appl. Math. Sci., 7 (142) (2013), 7093-7096.
6. [6] C. J. Harman, Complex Fibonacci numbers, The Fibonacci Quarterly, 19 (1) (1981), 82-86.
7. [7] R. R. de Oliveira, F. R. V. Alves, R. E. B. Paiva, Identidades bi e tridimensionais para os numeros de Fibonacci na forma complexa, C.Q.D.-Revista Eletrˆonica Paulista de Matem´atica, 11 (2) (2017), 91-106.
8. [8] R. R. de. Oliveira, Engenharia didatica sobre o modelo de complexificacao da sequencia generalizada de Fibonacci: Relacoes recorrentes n-dimensionais e representacoes polinomiais e matriciais. Dissertacao de Mestrado Academico do Programa de Pos-graduacao em Ensino de Ciencias e Matematica do Instituto Federal de Educacao, Ciencia e Tecnologia do Ceara - IFCE - Campus Fortaleza, 2018.

9. [9] R. P. M. Vieira, F. R. V. Alves, P. M. M. C. Catarino, Relacoes bidimensionais e identidades da sequencia de Leonardo, ReviSeM, 4 (2) (2019), 156-173.