THERMAL SCIENCE

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Yasemin Tasyurdu

GENERALIZED FIBONACCI NUMBERS WITH FIVE PARAMETERS

ABSTRACT
In this paper, we define five parameters generalization of Fibonacci numbers that generalizes Fibonacci, Pell, Modified Pell, Jacobsthal, Narayana, Padovan, k-Fibonacci, k-Pell, Modified k-Pell, k-Jacobsthal numbers and Fibonacci p-numbers, distance Fibonacci numbers, (2, k)-distance Fibonacci numbers, generalized (k, r)-Fibonacci numbers in the distance sense by extending the definition of a distance in the recurrence relation with two parameters and adding three parameters in the definition of this distance, simultaneously. Tiling and combinatorial interpretations of generalized Fibonacci numbers are presented, and explicit formulas that allow us to calculate the nth number are given. Also generating functions and some identities for these numbers are obtained.
KEYWORDS
combinatorial identities, distance Fibonacci numbers, tilings, generalized Fibonacci numbers, sums formulas
PAPER SUBMITTED: 2022-07-11
PAPER REVISED: 2022-08-30
PAPER ACCEPTED: 2022-10-05
PUBLISHED ONLINE: 2023-01-29
DOI REFERENCE: https://doi.org/10.2298/TSCI22S2495T
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2022, VOLUME 26, ISSUE Special issue 2, PAGES [495 - 505]
REFERENCES
  1. Horadam A. F., A Generalized Fibonacci Sequence, American Mathematical Monthly, 68 (1961), 5, pp. 455-459
  2. Horadam, A. F., Jacobsthal and Pell Curves, The Fibonacci Quarterly, 26 (1988), 1, pp. 79-83
  3. Allouche, J. P., Johnson, J., Narayana's Cows and Delayed Morphisms, Proceedings, 3rd Computer Music Conference JIM96, France, 1996
  4. Shannon, A. G., et al., Properties of Cordonnier, Perrin and Van der Laan Numbers, International Journal of Mathematical Education in Science and Technology, 37 (2006), 7, pp. 825-831
  5. Falcon, S. A., Plaza, A., On the Fibonacci k-Numbers, Chaos, Solitons & Fractals, 32 (2007), 5, pp. 1615-1624
  6. El-Mikkawy, M., Sogabe, T., A New Family of k-Fibonacci Numbers, Applied Mathematics and Computation, 215 (2010), 12 pp. 4456-4461
  7. Tasyurdu, Y., et al., On the a New Family of k-Fibonacci Numbers, Erzincan University Journal of Science and Technology, 9 (2016), 1, pp. 95-101
  8. Koshy, T., Fibonacci and Lucas Numbers with Applications, Wiley, New York, USA, 2001
  9. Koshy, T., Pell and Pell-Lucas Numbers with Applications, Springer, New York, USA, 2014
  10. Panwar, Y. K., A Note on the Generalized k-Fibonacci Sequence, MTU Journal of Engineering and Natural Sciences, 2 (2021), 2, pp. 29-39
  11. Ramirez, J. S., Sirvent V. F., A Note on the k-Narayana Sequence, Annales Mathematicae et Informaticae, 45 (2015), Jan., pp. 91-105
  12. Tasyurdu, Y., Generalized (p,q)-Fibonacci-Like Sequences and Their Properties, Journal of Mathematics Research, 11 (2019), 6, pp. 43-52
  13. Catarino, P., On some Identities and Generating Functions for k-Pell Numbers, Int. J. Math. Anal., 7 (2013), 38, pp. 1877-1884
  14. Catarino, P., Vasco, P., Modified k-Pell Sequence: Some Identities and Ordinary Generating Function, Applied Mathematical Sciences, 7 (2013), 121, pp. 6031-6037
  15. Jhala, D., et al., On some Identities for k-Jacobsthal Numbers, Int. J. Math. Anal., 7 (2013), 12, pp. 551-556
  16. Stakhov, A. P., Introduction into Algorithmic Measurement Theory, Soviet Radio, Moskow, Russia 1977
  17. Kwasnik, M., Wloch, I., The Total Number of Generalized Stable Sets and Kernels of Graphs, Ars Combin., 55 (2000), Apr., pp. 139-146
  18. Bednarz, U., et al., Distance Fibonacci Numbers, Their Interpretations and Matrix Generators, Commentat. Math., 53 (2013), 1, pp. 35-46
  19. Włoch, I., et al., On a New Type of Distance Fibonacci Numbers, Discrete Applied Mathematics, 161 (2013), 16-17, pp. 2695-2701
  20. Deveci, O., Karaduman, E., On the Padovan p-numbers, Hacettepe Journal of Mathematics and Statistics, 46 (2017), 4, pp. 579-592
  21. Falcon, S., Generalized (k, r)-Fibonacci Numbers, Gen. Math. Notes, 25 (2014), 2, pp. 148-158
  22. Bednarz, N., On (k, p)-Fibonacci numbers, Mathematics, 9 (2021), 727
  23. Brigham, R. C., et al., A Tiling Scheme for the Fibonacci Numbers, J. Recreational Math, 28 (1996-97), 1, pp. 10-17
  24. Benjamin, A. T., Quinn, J. J., Proofs that Really Count: The Art of Combinatorial Proof., Mathematical Association of America, Washington D. C.,2003
  25. Tasyurdu, Y., Cengiz, B., A Tiling Approach to Fibonacci p-numbers, Journal of Universal Mathematics, 5 (2022), 2, pp. 177-184
  26. Soykan Y., On generalized Padovan Numbers, On-line first, 10.20944/preprints202110. 0101.v1, 2021
  27. Kilic, E., The Binet Formula, Sums and Representations of Generalized Fibonacci p-numbers, Eur. J. Combin., 29 (2008), 3, pp. 701-711
  28. Kuhapatanakul, K. The Fibonacci p-numbers and Pascal's Triangle, Cogent Mathematics, 3 (2016), 1, 1264176
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Generalized Fibonacci numbers with five parameters

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