The Gaussian Sequence 3th Order Mod m

Year 2024, Volume: 7 Issue: 3 , 135 - 146 , 29.09.2024

Renata Vieira , Renata Teófilo De Sousa , Francisco Regis Alves

https://doi.org/10.33434/cams.1499700

https://izlik.org/JA25JB63DX

Abstract

The work addresses the study of third-order recurrent sequences for mod m cases. Thus, some definitions aim to transform infinite sequences into finite ones. In this regard, the Fourier transform is used as a visualization technique, explored in Google Colab. The mathematical theorems presented are established to examine the patterns of these sequences and their corresponding cycles. As a future perspective, it is intended to investigate other mathematical theorems to generalize the sequences into finite groups.

Keywords

Finite groups , Gaussian , Sequences 3th order

References

• [1] H. Aydin, R. Dikici, General Fibonacci sequences in finite groups, Fibonacci Quart., 36(3) (1998), 216-221.
• [2] B. Kuloglu, E. Ozkan, M. Marin, On the period of Pell-Narayana sequence in some groups, arxiv, (2023), 1-16.
• [3] E. Özkan, 3-step Fibonacci sequences in Nilpotent groups, Appl. Math. Comput., 144(2) (2003), 517-527.
• [4] E. Özkan, On general Fibonacci sequences in groups, Turkish J. Math., 27(4) (2003) 525-538.
• [5] D. Wall, Fibonacci series modulo m, The American Mathematical Monthly, 67(6) (1960), 525-532.
• [6] R. Vieira, E. Spreafico, F. Alves, P. Catarino, A note of the combinatorial interpretation of the Perrin and Tetrarrin sequence, J. Univers. Math., 7(1) (2024), 1-11.
• [7] R. Vieira, F. Alves, P. Catarino, A note on Leonardo’s combinatorial approach, J. of Instr. Math., 4(2) (2023), 119-126.
• [8] R. Vieira, F. Alves, P. Catarino, A didactic engineering for the study of the Padovan’s combinatory model, Pedagogical Research, 9(3) (2024), 1-9.
• [9] F. Alves, Uma discuss˜ao de Artigos envolvendo propriedades da sequencia de Fibonacci apoiada na tecnologia, Anais Do VI HTEM, (2013), 1-17.
• [10] P. Seenukul, S. Netmanee, T. Panyakhun, R. Auiseekaen, Sa-Muangchan, Matrices which have similar properties to Padovan Q-matrix and its generalized relations, Sakon Nakhon Rajabhat Univ. J. Sci. Tech., 7(2) (2015), 90-94.
• [11] R. Vieira, F. Alves, Explorando a sequencia de Padovan atraves de investigaao historica e abordagem epistemologica, Boletim GEPEM, 74 (2019), 162-169.
• [12] A. Shannon, P. Anderson, A. Horadam, Properties of cordonnier, Perrin and Van der Laan numbers, Internat. J. Math. Ed. Sci. Tech., 37(7) (2006), 825-831.
• [13] J. Shtayat, A. Al-Kaleeb, The Perrin R-matrix and more properties with an application, J. Discrete Math. Sci. Cryptogr., 25(4) (2022), 41-52.
• [14] J. Ramirez, V. Sirvent, A note on the k-Narayana sequence, Ann. Math. Inform., 45 (2015), 91-105.
• [15] Y. Soykan, On generalized Narayana numbers, Int. J. Adv. Appl. Math. Mech, 7(3) (2020), 43-56.
• [16] P. Catarino, A. Borges, On Leonardo numbers, Int. J. Adv. Appl. Math. Mech., 89(1) (2020), 75-86.
• [17] A. Shannon, O. Devici, A note on generalized and extended Leonardo sequences, Notes Number Theory Discrete Math., 28(1) (2022), 109-114.
• [18] S. Hulku, O. Erdag, O. Deveci, Complex-type Narayana sequence and its application, Maejo Int. J. Sci. Technol., 17(2) (2023) 163-176.
• [19] S. Knox, Fibonacci sequences in finite groups Fibonacci sequences in finite groups, Mathematical Sciences Technical Reports (MSTR), 142 (1990), 1-12.
• [20] B. Kuloglu, E. Ozkan, A. Shannon, The Narayana sequence in finite groups, Fibonacci Quarterly, 60(5) (2022), 212-221.
• [21] S. Tas, E. Karaduman, The Padovan sequences in finite groups, Chiang Mai J. Sci., 41(2) (2014), 456-462.
• [22] F. Alves, R. Vieira, P. Catarino, Visualizing the Newtons fractal from the recurring linear sequence with Google Colab: An example of Brazil X Portugal research, Int. Electron. J. Math. Ed., 15(3) (2020), 1-19.
• [23] M. Aschbacher, Finite Groups Theory, Cambridge University Press, 2000.
• [24] D. Collins, Generating Sequences of Finite Groups, Cornell University, 2009.
• [25] D. Tasci, Gaussian Padovan and Gaussian Pell-Padovan sequences, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat., 67(2) (2018), 82-88.
• [26] R. P. M. Vieira, Engenharia Didatica (ED): o caso da Generalizacao eComplexificacao da Sequencia de Padovan ou Cordonnier. 2020. 266f. Programa de Pos-Graduacao em Ensino de Ciencias e Matematica, Instituto Federal de Educacao, Ciencia e Tecnologia do Estado do Cear´a. Mestrado Acadˆemico em Ensino de Ciˆencias e Matem´atica, 2020.
• [27] O. Devici, E. Karaduman, G. Saglam. The Jacobsthal sequences in finite groups, Bull. Iranian Math. Soc., 42(1), 79-89, 2016.
• [28] M. Kartal, Gaussian Padovan and Gaussian Perrin numbers and properties of them, Asian-Eur. J. Math., 12(4) (2019), 1-8.
• [29] J. Jordan, Gaussian Fibonacci and Lucas numbers, The Fibonacci Quarterly, 3 (1965), 315-318.
• [30] E. Özkan, B. Kuloğlu, On The New Narayana Polynomials, The Gauss Narayana Numbers and their polynomials, Asian-Eur. J. Math., 14(6) (2021), 1-15.
• [31] Y. Soykan, M. Göcen, S. Çevikel, On matrix sequences of Narayana and Narayana-Lucas numbers, Karaelmas Sci. Engrg. J., 11(1) (2021), 83-90.
• [32] D. Tasci, On Gaussian Leonardo numbers, Contrib. Math., 7 (2023), 34-40.