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Journals / Journal of Universal Mathematics / 2022 / Cilt: 5 Sayı: 2

A TILING APPROACH TO FIBONACCI p-NUMBERS

Yasemin Taşyurdu(Erzincan Binali Yıldırım Üniversitesi)0000-0002-9011-8269
Berke Cengiz(ERZİNCAN BİNALİ YILDIRIM ÜNİVERSİTESİ)0000-0001-8372-3332
Pages
177–184
Publication Date
2022-07-31
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AI Summary & Key Takeaways

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

Moreover, the product identities and sum formulas of these numbers with special subscripts are given by tiling interpretations that allow the derivation of their properties.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:In this paper, we introduce tiling representations of Fibonacci p-numbers, which are generalizations of the well-known Fibonacci and Narayana numbers, and generalized in the distance sense.
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Key Findings & Data:We obtain Fibonacci p-numbers count the number of distinct ways to tile a 1 × n board using various 1 × r, r-ominoes from r = 1 up to r = p + 1.
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Conclusion & Impact:Moreover, the product identities and sum formulas of these numbers with special subscripts are given by tiling interpretations that allow the derivation of their properties.
🔬Design:Araştırma Makalesi
📍Scope:Comprehensive Sample
⚙️Variables:Fibonacci p-Numbers · Generalized Fibonacci Numbers · Sum Formulas

Abstract

In this paper, we introduce tiling representations of Fibonacci p-numbers, which are generalizations of the well-known Fibonacci and Narayana numbers, and generalized in the distance sense. We obtain Fibonacci p-numbers count the number of distinct ways to tile a 1 × n board using various 1 × r, r-ominoes from r = 1 up to r = p + 1. Moreover, the product identities and sum formulas of these numbers with special subscripts are given by tiling interpretations that allow the derivation of their properties.

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Cite This Article

Taşyurdu, Y., Cengiz, B. (2022). A TILING APPROACH TO FIBONACCI p-NUMBERS. Journal of Universal Mathematics, Cilt: 5 Sayı: 2, 177–184. https://doi.org/10.33773/jum.1142766

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References (21)

  1. S. Falcon and A. Plaza, On the Fibonacci k-Numbers, Solitons-Fractals. Vol. 32, N. 5, pp. 1615-24 (2007).
  2. M. El-Mikkawy and T. Sogabe, A New Family of k-Fibonacci Numbers, Applied Mathematics and Computation, Vol. 215, pp.4456–4461 (2010).
  3. Y. Tasyurdu and N. Cobanoglu 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).
  4. J.P. Allouche and J. Johnson, Narayanas Cows and Delayed Morphisms, Articles of 3rd Computer Music Conference JIM96 (1996).
  5. A.P. Stakhov, Introduction into Algorithmic Measurement Theory, Soviet Radio, Moskow, (1977).
  6. A.P. Stakhov, Fibonacci Matrices A Generalization of the Cassini Formula and A New Coding Theory, Solitons and Fractals, Vol. 30, pp. 56–66 (2006).
  7. A. Stakhov and B. Rozin, Theory of Binet Formulas for Fibonacci and Lucas p-Numbers, Solitons and Fractals, Vol. 27, N. 5, pp. 1163–1177 (2006).
  8. E. Kilic, The Binet Formula Sums and Representations of Generalized Fibonacci p-numbers, Eur. J. Combin, Vol. 29, pp.701–711 (2008).
  9. K. Kuhapatanakul, The Fibonacci p-Numbers and Pascals Triangle, Cogent Mathematics, Vol. 3, N. 1, pp.7 (2016).
  10. Y.K.A. Panwar, Note on the Generalized k-Fibonacci Sequence, MTU Journal of Engineering and Natural Sciences, Vol. 2, N.2, pp. 29-39 (2021).
  11. T.Koshy, Fibonacci and Lucas Numbers with Applications, Wiley, New York NY USA, (2001).
  12. Y. Tasyurdu, Generalized (p, q)-Fibonacci-Like Sequences and Their Properties, Journal of Mathematics Research, Vol. 11, N. 6, pp. 43-52 (2019).
  13. M. Kwasnik and I. W loch, The Total Number of Generalized Stable Sets and Kernels of Graphs, Ars Combin, Vol. 55, pp. 139–146 (2000).
  14. U. Bednarz, A. W loch and M. Wo lowiec-Musia, Distance Fibonacci Numbers Their Interpretations and Matrix Generators, Commentat. Math, Vol. 53, N. 1, pp. 35–46 (2013).
  15. I. W loch and U. Bednarz and D. Brod and A. W loch and M. Wo lowiec-Musia, One New Type of Distance Fibonacci numbers, Discrete Applied Mathematics, Vol. 161, N. (16-17), pp. 2695-2701 (2013).
  16. A.B. Brother, Fibonacci. Numbers and Geometry, The Fibonacci Quarterly, Vol. 10, N. 3, pp. 303-318 (1972).
  17. H.L. Holden, Fibonacci Tiles, The Fibonacci Quarterly, Vol. 13, N. 1, pp. 45-49 (1975).
  18. E. Verner and J.R. Hoggatt and K. Alladi, Generalized Fibonacci Tiling, Fibonacci Quarterly, Vol. 13 N. 2, pp. 137-149(1974).
  19. 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).
  20. A.T. Benjamin and J.J. Quinn, Proofs That Really Count The Art of Combinatorial, Proof Mathematical Association of America, (2003).
  21. A.T. Benjamin and J.J. Quinn, The Fibonacci Numbers Exposed More Discretely, Math. Magazine, Vol. 33, pp. 182–192 (2002).