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

Year 2022, , 81 - 87, 31.07.2022
https://doi.org/10.33773/jum.1142805

Abstract

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.

References

  • [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).
Year 2022, , 81 - 87, 31.07.2022
https://doi.org/10.33773/jum.1142805

Abstract

References

  • [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).
There are 14 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Yasemin Taşyurdu 0000-0002-9011-8269

Naime Şeyda Türkoğlu 0000-0003-2301-3958

Publication Date July 31, 2022
Submission Date July 9, 2022
Acceptance Date July 30, 2022
Published in Issue Year 2022

Cite

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. July 2022;5(2):81-87. doi:10.33773/jum.1142805
Chicago Taşyurdu, Yasemin, and Naime Şeyda Türkoğlu. “A TILING INTERPRETATION FOR (p,q)-FIBONACCI AND (p,q)-LUCAS NUMBERS”. Journal of Universal Mathematics 5, no. 2 (July 2022): 81-87. https://doi.org/10.33773/jum.1142805.
EndNote Taşyurdu Y, Türkoğlu NŞ (July 1, 2022) A TILING INTERPRETATION FOR (p,q)-FIBONACCI AND (p,q)-LUCAS NUMBERS. Journal of Universal Mathematics 5 2 81–87.
IEEE Y. Taşyurdu and N. Ş. Türkoğlu, “A TILING INTERPRETATION FOR (p,q)-FIBONACCI AND (p,q)-LUCAS NUMBERS”, JUM, vol. 5, no. 2, pp. 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 (July 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 and Naime Şeyda Türkoğlu. “A TILING INTERPRETATION FOR (p,q)-FIBONACCI AND (p,q)-LUCAS NUMBERS”. Journal of Universal Mathematics, vol. 5, no. 2, 2022, pp. 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.