The Gaussian Sequence 3th Order Mod m
Yıl 2024,
Cilt: 7 Sayı: 3, 135 - 146, 29.09.2024
Renata Vieira
,
Renata Teófilo De Sousa
,
Francisco Regis Alves
Öz
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.
Kaynakça
- [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.
The Gaussian Sequence 3th Order Mod m
Yıl 2024,
Cilt: 7 Sayı: 3, 135 - 146, 29.09.2024
Renata Vieira
,
Renata Teófilo De Sousa
,
Francisco Regis Alves
Öz
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.
Kaynakça
- [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.