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Journals / Karaelmas Fen ve Mühendislik Dergisi / 2016 / Cilt: 6 - Sayı: 1

Padovan-Type Sequences via Hurwitz Matrices

Hurwitz Matrisleri Yardımıyla Elde Edilen Padovan-Tipli Diziler

Ömür DEVECİ
Pages
1–8
Publication Date
2016-01-01
DOI
—
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Finally, we obtain the lengths of the periods of the extended sequences in the semidihedral group 2mSD and the modular maximal-cyclic group M_m 2 for m>=4 as applications of the results obtained.

📌 3 Key Takeaways

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Objective & Hypothesis:In this paper, we define the recurrence sequences by using the Hurwitz matrices which are obtained from the characteristic polynomials of the Padovan, the Pell-Padovan and the Jacobsthal-Padovan sequences and then, we obtain miscellaneous properties of these sequences.
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Key Findings & Data:Also, we study these sequences modulo m and we obtain the cyclic groups which are generated by the generating matrices when read modulo m.
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Conclusion & Impact:Finally, we obtain the lengths of the periods of the extended sequences in the semidihedral group 2mSD and the modular maximal-cyclic group M_m 2 for m>=4 as applications of the results obtained.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Hurwitz matrisi · Dizi · Grup

Abstract

In this paper, we define the recurrence sequences by using the Hurwitz matrices which are obtained from the characteristic polynomials of the Padovan, the Pell-Padovan and the Jacobsthal-Padovan sequences and then, we obtain miscellaneous properties of these sequences. Also, we study these sequences modulo m and we obtain the cyclic groups which are generated by the generating matrices when read modulo m. Then we derive the relationships among the orders of the obtained cyclic groups and the periods of the defined sequences according to modulo m. Furthermore, we extend the PadovanHurwitz, the Pell-Padovan-Hurwitz sequences to groups. Finally, we obtain the lengths of the periods of the extended sequences in the semidihedral group 2mSD and the modular maximal-cyclic group M_m 2 for m>=4 as applications of the results obtained.

Özet

Bu makalede, Padovan, Pell-Padovan ve Jacobsthal-Padovan dizilerinin karakteristik polinomlarından elde edilen Hurwitz matrisleri kullanılarak indirgemeli dizler tanımlanmış ve bu dizilerin çeşitli özellikleri verilmiştir. Ayrıca, bu diziler m modülünde çalışılmış ve dizilerin üreteç matrisleri m modülüne indirgenerek devirli grupların üreteçleri olarak kabul edilip devirli gruplar elde edilmiştir. Bunun sonucu olarak, elde edilen devirli grupların mertebeleri ile tanımlanan dizilerin m modülüne göre periyotları arasında bağıntılar üretilmiştir. Buna ek olarak, Padovan-Hurwitz ve Pell-Padovan-Hurwitz dizileri gruplara genişletilmiştir. En sonunda, elde edilen sonuçların uygulaması olarak, m ≥ 4 için SD2m semidihedral grup ve M_m 2 modular maximal-cyclic grubun genişletilmiş dizilerinin periyotlarının uzunlukları elde edilmişitir

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Cite This Article

DEVECİ, Ö. (2016). Padovan-Type Sequences via Hurwitz Matrices. Karaelmas Fen ve Mühendislik Dergisi, Cilt: 6 - Sayı: 1, 1–8. https://ajindex.com/articles/684050

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References (20)

  1. Aydin, H., Smith, GC. 1994. Finite p-quotients of some cyclically presented groups, J. London Math. Soc., 49: 83-92.
  2. Campbell, CM., Campbell, PP. 2009. The Fibonacci lengths of binary polyhedral groups and related groups. Congr. Numer., 194: 95-102.
  3. Coxeter, HSM., Greitzer, SL., 1967.
  4. Geometry revisited, Washington, DC: Math. Assoc. Amer., 41 pp. Deveci, O., 2015. The Pell-Padovan sequences and the Jacobsthal- Padovan sequences in finite groups, Util. Math., 98: 257-270.
  5. Deveci, O., Akuzum, Y., 2015. The recurrence sequences via Hurwitz matrices, Sci. Ann. “Al. I. Cuza”. Univ. Iasi, (in press). Deveci, O., Akuzum, Y., 2014. The cycylic groups via MacWilliams and Chebyshev matrices, J. Math. Reseacrh, 6: 55-58.
  6. Deveci, O., Avci, M., 2015.
  7. Fibonacci p-sequences in groups, Maejo Int. J. Sci. Tech., 9: 301-311.
  8. Deveci, O., Karaduman, E., 2012. The cyclic groups via the Pascal matrices and the generalized Pascal matrices, Linear Algebra Appl., 437: 2538-2545.
  9. Doostie, H., Hashemi, M., 2006. Fibonacci lengths involving the Wall number k(n), J. Appl. Math. Comput., 20: 171-180.
  10. Dikici, R., Smith, GC., 1997. Fibonacci sequences in finite nilpotent groups, Turkish J. Math., 21: 133-142.
  11. Dummit, DS., Foote, R., 2004. Abstract algebra 3st Edn., Wiley, 71 pp. Falcon, S., Plaza, A., 2009. k-Fibonacci sequences modulo m, Chaos Solitons Fractals, 41: 497-504.
  12. Frey, DD., Sellers, JA., 2000. Jacobsthal numbers and alternating sign matrices, J. Integer Seq., 3: Article 00.2.3. Ozkan, E., Aydin, H., Dikici, R., 2003. 3-step Fibonacci series modulo m, Appl. Math. Comput., 143: 165-172.
  13. Shannon, AG., Anderson, PG., Horadam, AF., 2006a. Properties of cordonnier Perrin and Van der Lan numbers, Int. J. Math. Educ. Sci. Technol., 37: 825-831.
  14. Shannon, AG., Horadam, AF., Anderson, PG., 2006b. The auxiliary equation associated with plastic number, Notes Number Theory Disc. Math., 12: 1-12.
  15. Stakhov, AP., Rozin, B., 2006. Theory of Binet formulas for Fibonacci and Lucas p-numbers, Chaos Solitons Fractals, 27: 1162-1177.
  16. Tas, S., Karaduman, E., 2014. The Padovan sequences in finite groups, Chaing Mai J. Sci., 41: 456-462.
  17. Wall, DD., 1960.
  18. Fibonacci series modulo m, Amer. Math. Monthly, 67: 525-532.
  19. Yilmaz, F., Bozkurt, D., 2009. The generalized order-k Jacobsthal numbers, Int. J. Contemp. Math. Sci., 4: 1685-1694.
  20. Lü, K., Wang, J., 2007. k-step Fibonacci sequence modulo m, Util. Math., 71: 169-178.