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

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.

Keywords:

Fibonacci Number, Generalized Fibonacci Numbers Lucas Number, Tilings,

PDF

___

[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).