Araştırma Makalesi
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A TILING INTERPRETATION FOR (p,q)-FIBONACCI AND (p,q)-LUCAS NUMBERS

Yıl 2022, Cilt: 5 Sayı: 2, 81 - 87, 31.07.2022
https://doi.org/10.33773/jum.1142805

Öz

In this paper, we introduce a tiling approach to (p,q)-Fibonacci and (p,q)-Lucas numbers that generalize of the well-known Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal ve Jacobsthal-Lucas numbers. We show that nth (p,q)-Fibonacci number is interpreted as the number of ways to tile a 1×n board with cells labeled 1,2,...,n using colored 1×1 squares and 1×2 dominoes, where there are p kind colors for squares and q kind colors for dominoes. Then nth (p,q)-Lucas number is interpreted as the number of ways to tile a circular 1×n board with squares and dominoes. We also present some generalized Fibonacci and Lucas identities using this tiling approach.

Kaynakça

  • [1] A.F. Horadam, A Generalized Fibonacci Sequence, American Mathematical Monthly, Vol. 68, N. 5, pp. 455-459 (1961).
  • [2] S. Falcon and A. Plaza, On the Fibonacci k-Numbers, Solitons - Fractals. Vol. 32, N. 5, pp. 1615-1624 (2007).
  • [3] S. Falcon, On the k-Lucas numbers, International Journal of Contemporary Mathematical Sciences, Vol. 6, N. 21, pp. 1039-1050 (2011).
  • [4] M. El-Mikkawy and T. Sogabe, A New Family of k-Fibonacci Numbers, Applied Mathematics and Computation, Vol. 215, pp.4456–4461 (2010).
  • [5] Y. Taşyurdu and N. Cobanoğlu and Z. Dilmen, On The A New Family of k-Fibonacci Numbers, Erzincan University Journal of Science and Technology. Vol. 9, N. 1, pp. 95-101 (2016).
  • [6] Y.K. Panwar, A Note On The Generalized k-Fibonacci Sequence, MTU Journal of Engineering and Natural Sciences, Vol. 2, N. 2, pp. 29-39 (2021).
  • [7] A. Suvarnamani and M. Tatong, Some Properties of (p, q)-Fibonacci Numbers, Science and Technology RMUTT Journal, Vol. 5, N. 2, pp. 17–21 (2015).
  • [8] A. Suvarnamani, Some Properties of (p, q)-Lucas Number, Kyungpook Mathematical Journal, Vol. 56, pp. 43-52 (2016).
  • [9] Y. Taşyurdu, Generalized (p, q)-Fibonacci-Like Sequences and Their Properties, Journal of Mathematics Research, Vol. 11, N. 6, pp. 367-370 (2019).
  • [10] T.Koshy, Fibonacci and Lucas Numbers with Applications, Wiley, New York USA, (2001).
  • [11] B.A. Brousseau, Fibonacci Numbers and Geometry, The Fibonacci Quarterly, Vol. 10, N. 3, pp. 303-318 (1972).
  • [12] R.C. Brigham and R.M. Caron and P.Z. Chinn and R.P. Grimaldi, A Tiling Scheme for the Fibonacci Numbers, J. Recreational Math, Vol. 28, N. 1, pp. 10–17 (1996-97).
  • [13] A.T. Benjamin and J.J. Quinn, The Fibonacci Numbers Exposed More Discretely, Math. Magazine, Vol. 33, pp. 182–192 (2002).
  • [14] A.T. Benjamin and J.J. Quinn, Proofs That Really Count The Art of Combinatorial, Proof Mathematical Association of America, (2003).
Yıl 2022, Cilt: 5 Sayı: 2, 81 - 87, 31.07.2022
https://doi.org/10.33773/jum.1142805

Öz

Kaynakça

  • [1] A.F. Horadam, A Generalized Fibonacci Sequence, American Mathematical Monthly, Vol. 68, N. 5, pp. 455-459 (1961).
  • [2] S. Falcon and A. Plaza, On the Fibonacci k-Numbers, Solitons - Fractals. Vol. 32, N. 5, pp. 1615-1624 (2007).
  • [3] S. Falcon, On the k-Lucas numbers, International Journal of Contemporary Mathematical Sciences, Vol. 6, N. 21, pp. 1039-1050 (2011).
  • [4] M. El-Mikkawy and T. Sogabe, A New Family of k-Fibonacci Numbers, Applied Mathematics and Computation, Vol. 215, pp.4456–4461 (2010).
  • [5] Y. Taşyurdu and N. Cobanoğlu and Z. Dilmen, On The A New Family of k-Fibonacci Numbers, Erzincan University Journal of Science and Technology. Vol. 9, N. 1, pp. 95-101 (2016).
  • [6] Y.K. Panwar, A Note On The Generalized k-Fibonacci Sequence, MTU Journal of Engineering and Natural Sciences, Vol. 2, N. 2, pp. 29-39 (2021).
  • [7] A. Suvarnamani and M. Tatong, Some Properties of (p, q)-Fibonacci Numbers, Science and Technology RMUTT Journal, Vol. 5, N. 2, pp. 17–21 (2015).
  • [8] A. Suvarnamani, Some Properties of (p, q)-Lucas Number, Kyungpook Mathematical Journal, Vol. 56, pp. 43-52 (2016).
  • [9] Y. Taşyurdu, Generalized (p, q)-Fibonacci-Like Sequences and Their Properties, Journal of Mathematics Research, Vol. 11, N. 6, pp. 367-370 (2019).
  • [10] T.Koshy, Fibonacci and Lucas Numbers with Applications, Wiley, New York USA, (2001).
  • [11] B.A. Brousseau, Fibonacci Numbers and Geometry, The Fibonacci Quarterly, Vol. 10, N. 3, pp. 303-318 (1972).
  • [12] R.C. Brigham and R.M. Caron and P.Z. Chinn and R.P. Grimaldi, A Tiling Scheme for the Fibonacci Numbers, J. Recreational Math, Vol. 28, N. 1, pp. 10–17 (1996-97).
  • [13] A.T. Benjamin and J.J. Quinn, The Fibonacci Numbers Exposed More Discretely, Math. Magazine, Vol. 33, pp. 182–192 (2002).
  • [14] A.T. Benjamin and J.J. Quinn, Proofs That Really Count The Art of Combinatorial, Proof Mathematical Association of America, (2003).
Toplam 14 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Araştırma Makalesi
Yazarlar

Yasemin Taşyurdu 0000-0002-9011-8269

Naime Şeyda Türkoğlu 0000-0003-2301-3958

Yayımlanma Tarihi 31 Temmuz 2022
Gönderilme Tarihi 9 Temmuz 2022
Kabul Tarihi 30 Temmuz 2022
Yayımlandığı Sayı Yıl 2022 Cilt: 5 Sayı: 2

Kaynak Göster

APA Taşyurdu, Y., & Türkoğlu, N. Ş. (2022). A TILING INTERPRETATION FOR (p,q)-FIBONACCI AND (p,q)-LUCAS NUMBERS. Journal of Universal Mathematics, 5(2), 81-87. https://doi.org/10.33773/jum.1142805
AMA Taşyurdu Y, Türkoğlu NŞ. A TILING INTERPRETATION FOR (p,q)-FIBONACCI AND (p,q)-LUCAS NUMBERS. JUM. Temmuz 2022;5(2):81-87. doi:10.33773/jum.1142805
Chicago Taşyurdu, Yasemin, ve Naime Şeyda Türkoğlu. “A TILING INTERPRETATION FOR (p,q)-FIBONACCI AND (p,q)-LUCAS NUMBERS”. Journal of Universal Mathematics 5, sy. 2 (Temmuz 2022): 81-87. https://doi.org/10.33773/jum.1142805.
EndNote Taşyurdu Y, Türkoğlu NŞ (01 Temmuz 2022) A TILING INTERPRETATION FOR (p,q)-FIBONACCI AND (p,q)-LUCAS NUMBERS. Journal of Universal Mathematics 5 2 81–87.
IEEE Y. Taşyurdu ve N. Ş. Türkoğlu, “A TILING INTERPRETATION FOR (p,q)-FIBONACCI AND (p,q)-LUCAS NUMBERS”, JUM, c. 5, sy. 2, ss. 81–87, 2022, doi: 10.33773/jum.1142805.
ISNAD Taşyurdu, Yasemin - Türkoğlu, Naime Şeyda. “A TILING INTERPRETATION FOR (p,q)-FIBONACCI AND (p,q)-LUCAS NUMBERS”. Journal of Universal Mathematics 5/2 (Temmuz 2022), 81-87. https://doi.org/10.33773/jum.1142805.
JAMA Taşyurdu Y, Türkoğlu NŞ. A TILING INTERPRETATION FOR (p,q)-FIBONACCI AND (p,q)-LUCAS NUMBERS. JUM. 2022;5:81–87.
MLA Taşyurdu, Yasemin ve Naime Şeyda Türkoğlu. “A TILING INTERPRETATION FOR (p,q)-FIBONACCI AND (p,q)-LUCAS NUMBERS”. Journal of Universal Mathematics, c. 5, sy. 2, 2022, ss. 81-87, doi:10.33773/jum.1142805.
Vancouver Taşyurdu Y, Türkoğlu NŞ. A TILING INTERPRETATION FOR (p,q)-FIBONACCI AND (p,q)-LUCAS NUMBERS. JUM. 2022;5(2):81-7.