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On The Connections Between Jacobsthal Numbers and Fibonacci p-Numbers

Year 2020, Volume: 5 Issue: 2, 147 - 156, 31.10.2020

Abstract

In this paper, we define the Fibonacci-Jacobsthal p-sequence and then we discuss the connection of the Fibonacci-Jacobsthal p-sequence with Jacobsthal and Fibonacci p-sequences. Also, we provide a new Binet formula and a new combinatorial representation of Fibonacci-Jacobsthal p-numbers by the aid of the nth power of the generating matrix the Fibonacci-Jacobsthal p-sequence. Furthermore, we derive relationships between the Fibonacci-Jacobsthal p-numbers and their permanent, determinant and sums of certain matrices.

References

  • Bradie B. Extension and refinements of some properties of sums involving Pell number. Missouri J.Math. Sci. 22(1), 2010, 37–43.
  • Brualdi RA, Gibson PM. Convex polyhedra of doubly stochastic matrices I: applications of permanent function. J. Combin. Theory, Series A. 22(2), 1997, 194–230.
  • Chen WYC, Louck JD. The combinatorial power of the companion matrix. Linear Algebra Appl. 232, 1996, 261–278.
  • Devaney R. The Mandelbrot set and the Farey tree, and the Fibonacci sequence. Amer. Math. Monthly. 106, 1999, 289–302.
  • Deveci O. The Jacobsthal-Padovan p-sequences and their applications. Proc. Rom. Acad. Series A. 20(3), 2019, 215–224.
  • Erdag O, Deveci O, Shannon AG. Matrix Manipulations for Properties of Jacobsthal p-Numbers and their Generalizations. The Scientific Annals of “Al. I. Cuza” University of Iasi. in press.
  • Deveci O, Adiguzel Z, Akuzum Y. On the Jacobsthal-circulant-Hurwitz numbers. Maejo International Journal of Science and Technology. 14(1), 2020, 56–67.
  • Frey DD, Sellers JA. Jacobsthal numbers and alternating sign matrices. J. Integer Seq. 3, 2000, Article 00.2.3.
  • Gogin N, Myllari AA. The Fibonacci-Padovan sequence and MacWilliams transform matrices. Programing and Computer Software, published in Programmirovanie. 33(2), 2007, 74–79.
  • Horadam AF. Jacobsthal representations numbers. Fibonacci Quart. 34, 1996, 40–54.
  • Johnson B. Fibonacci identities by matrix methods and generalisation to related sequences. http://maths.dur.ac.uk/˜dma0rcj/PED/fib.pdf, March 25, 2003.
  • Kalman D. Generalized Fibonacci numbers by matrix methods. Fibonacci Quart. 20(1), 1982, 73–76.
  • Kilic E. The Binet fomula, sums and representations of generalized Fibonacci p-numbers. European Journal of Combinatorics. 29, 2008, 701–711.
  • Kilic E, Tasci D. The generalized Binet formula, representation and sums of the generalized order-k Pell numbers. Taiwanese J. Math. 10(6), 2006, 1661–1670.
  • Kocer EG. The Binet formulas for the Pell and Pell-Lucas p-numbers. Ars Comb. 85, 2007, 3–17.
  • Koken F, Bozkurt D. On the Jacobsthal numbers by matrix methods. Int. J. Contemp. Math. Sciences. 3(13), 2008, 605–614.
  • Lancaster P, Tismenetsky M. The theory of matrices: with applications. Elsevier. 1985.
  • Lidl R, Niederreiter H. Introduction to finite fields and their applications. Cambridge UP. 1986.
  • Shannon AG, Anderson PG, Horadam AF. Properties of cordonnier Perrin and Van der Lan numbers. Internat. J. Math. Ed. Sci. Tech. 37(7), 2006, 825–831.
  • Shannon AG, Horadam AF, Anderson PG. The auxiliary equation associated with the plastic number. Notes Number Theory Discrete Math. 12(1), 2006, 1–12.
  • Stakhov AP. A generalization of the Fibonacci Q-matrix. Rep. Natl. Acad. Sci. Ukraine. 9, 1999, 46–49.
  • Stakhov AP, Rozin B. Theory of Binet formulas for Fibonacci and Lucas p-numbers. Chaos, Solitions Fractals. 27, 2006, 1162–1177.
  • Tasci D, Firengiz MC. Incomplete Fibonacci and Lucas p-numbers. Math. Comput. Modell. 52, 2010, 1763–1770.
Year 2020, Volume: 5 Issue: 2, 147 - 156, 31.10.2020

Abstract

References

  • Bradie B. Extension and refinements of some properties of sums involving Pell number. Missouri J.Math. Sci. 22(1), 2010, 37–43.
  • Brualdi RA, Gibson PM. Convex polyhedra of doubly stochastic matrices I: applications of permanent function. J. Combin. Theory, Series A. 22(2), 1997, 194–230.
  • Chen WYC, Louck JD. The combinatorial power of the companion matrix. Linear Algebra Appl. 232, 1996, 261–278.
  • Devaney R. The Mandelbrot set and the Farey tree, and the Fibonacci sequence. Amer. Math. Monthly. 106, 1999, 289–302.
  • Deveci O. The Jacobsthal-Padovan p-sequences and their applications. Proc. Rom. Acad. Series A. 20(3), 2019, 215–224.
  • Erdag O, Deveci O, Shannon AG. Matrix Manipulations for Properties of Jacobsthal p-Numbers and their Generalizations. The Scientific Annals of “Al. I. Cuza” University of Iasi. in press.
  • Deveci O, Adiguzel Z, Akuzum Y. On the Jacobsthal-circulant-Hurwitz numbers. Maejo International Journal of Science and Technology. 14(1), 2020, 56–67.
  • Frey DD, Sellers JA. Jacobsthal numbers and alternating sign matrices. J. Integer Seq. 3, 2000, Article 00.2.3.
  • Gogin N, Myllari AA. The Fibonacci-Padovan sequence and MacWilliams transform matrices. Programing and Computer Software, published in Programmirovanie. 33(2), 2007, 74–79.
  • Horadam AF. Jacobsthal representations numbers. Fibonacci Quart. 34, 1996, 40–54.
  • Johnson B. Fibonacci identities by matrix methods and generalisation to related sequences. http://maths.dur.ac.uk/˜dma0rcj/PED/fib.pdf, March 25, 2003.
  • Kalman D. Generalized Fibonacci numbers by matrix methods. Fibonacci Quart. 20(1), 1982, 73–76.
  • Kilic E. The Binet fomula, sums and representations of generalized Fibonacci p-numbers. European Journal of Combinatorics. 29, 2008, 701–711.
  • Kilic E, Tasci D. The generalized Binet formula, representation and sums of the generalized order-k Pell numbers. Taiwanese J. Math. 10(6), 2006, 1661–1670.
  • Kocer EG. The Binet formulas for the Pell and Pell-Lucas p-numbers. Ars Comb. 85, 2007, 3–17.
  • Koken F, Bozkurt D. On the Jacobsthal numbers by matrix methods. Int. J. Contemp. Math. Sciences. 3(13), 2008, 605–614.
  • Lancaster P, Tismenetsky M. The theory of matrices: with applications. Elsevier. 1985.
  • Lidl R, Niederreiter H. Introduction to finite fields and their applications. Cambridge UP. 1986.
  • Shannon AG, Anderson PG, Horadam AF. Properties of cordonnier Perrin and Van der Lan numbers. Internat. J. Math. Ed. Sci. Tech. 37(7), 2006, 825–831.
  • Shannon AG, Horadam AF, Anderson PG. The auxiliary equation associated with the plastic number. Notes Number Theory Discrete Math. 12(1), 2006, 1–12.
  • Stakhov AP. A generalization of the Fibonacci Q-matrix. Rep. Natl. Acad. Sci. Ukraine. 9, 1999, 46–49.
  • Stakhov AP, Rozin B. Theory of Binet formulas for Fibonacci and Lucas p-numbers. Chaos, Solitions Fractals. 27, 2006, 1162–1177.
  • Tasci D, Firengiz MC. Incomplete Fibonacci and Lucas p-numbers. Math. Comput. Modell. 52, 2010, 1763–1770.
There are 23 citations in total.

Details

Primary Language English
Journal Section Volume V Issue II 2020
Authors

Özgür ERDAĞ

Ömür DEVECİ

Publication Date October 31, 2020
Published in Issue Year 2020 Volume: 5 Issue: 2

Cite

APA ERDAĞ, Ö., & DEVECİ, Ö. (2020). On The Connections Between Jacobsthal Numbers and Fibonacci p-Numbers. Turkish Journal of Science, 5(2), 147-156.
AMA ERDAĞ Ö, DEVECİ Ö. On The Connections Between Jacobsthal Numbers and Fibonacci p-Numbers. TJOS. October 2020;5(2):147-156.
Chicago ERDAĞ, Özgür, and Ömür DEVECİ. “On The Connections Between Jacobsthal Numbers and Fibonacci P-Numbers”. Turkish Journal of Science 5, no. 2 (October 2020): 147-56.
EndNote ERDAĞ Ö, DEVECİ Ö (October 1, 2020) On The Connections Between Jacobsthal Numbers and Fibonacci p-Numbers. Turkish Journal of Science 5 2 147–156.
IEEE Ö. ERDAĞ and Ö. DEVECİ, “On The Connections Between Jacobsthal Numbers and Fibonacci p-Numbers”, TJOS, vol. 5, no. 2, pp. 147–156, 2020.
ISNAD ERDAĞ, Özgür - DEVECİ, Ömür. “On The Connections Between Jacobsthal Numbers and Fibonacci P-Numbers”. Turkish Journal of Science 5/2 (October 2020), 147-156.
JAMA ERDAĞ Ö, DEVECİ Ö. On The Connections Between Jacobsthal Numbers and Fibonacci p-Numbers. TJOS. 2020;5:147–156.
MLA ERDAĞ, Özgür and Ömür DEVECİ. “On The Connections Between Jacobsthal Numbers and Fibonacci P-Numbers”. Turkish Journal of Science, vol. 5, no. 2, 2020, pp. 147-56.
Vancouver ERDAĞ Ö, DEVECİ Ö. On The Connections Between Jacobsthal Numbers and Fibonacci p-Numbers. TJOS. 2020;5(2):147-56.