Combinatorial proofs of identities for the generalized Leonardo numbers

Mark Shattuck
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 28, 2022, Number 4, Pages 778–790
DOI: 10.7546/nntdm.2022.28.4.778-790
Full paper (PDF, 210 Kb)

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Authors and affiliations

Mark Shattuck
Department of Mathematics, University of Tennessee
Knoxville, TN 37996, USA

Abstract

In this paper, we provide combinatorial proofs of several prior identities satisfied by the recently introduced generalized Leonardo numbers, denoted by , as well as derive some new formulas. To do so, we interpret as the enumerator of two classes of linear colored tilings of length . A comparable treatment is also given for the incomplete generalized Leonardo numbers. Finally, a -generalization of is obtained by considering the joint distribution of a pair of statistics on one of the aforementioned classes of colored tilings.

Keywords

• Leonardo number
• Fibonacci number
• Linear tiling
• Combinatorial proof

2020 Mathematics Subject Classification

• 05A19
• 11B39

References

1. Alp, Y., & Kocer, E. G. (2021). Some properties of Leonardo numbers. Konuralp Journal of Mathematics, 9(1), 183–189.
2. Benjamin, A. T., & Quinn, J. J. (2003). Proofs that Really Count: The Art of Combinatorial Proof. Mathematical Association of America, Washington, DC.
3. Catarino, P., & Borges, A. (2020). On Leonardo numbers. Acta Mathematica Universitatis Comenianae, 89(1), 75–86.
4. Catarino, P., & Borges, A. (2020). A note on incomplete Leonardo numbers. Integers, 20, #A43.
5. Dijkstra, E. W. (1981). Smoothsort–an alternative to sorting in situ. EWD-796a. Available online at: https://www.cs.utexas.edu/users/EWD/transcriptions/
   EWD07xx/EWD796a.html.
6. Dijkstra, E. W. (1981). Fibonacci numbers and Leonardo numbers. EWD-797. Available online at: https://www.cs.utexas.edu/users/EWD/transcriptions/
   EWD07xx/EWD797.html.
7. Karataş, A. (2022). On complex Leonardo numbers. Notes on Number Theory and Discrete Mathematics, 28(3), 458–465.
8. Kuhapatanakul, K., & Chobsorn, J. (2022). On the generalized Leonardo numbers. Integers, 22, #A48.
9. Kürüz, F., Dağdeviren, A., & Catarino, P. (2021). On Leonardo Pisano hybrinomials. Mathematics, 9, #2923.
10. Shannon, A. G. (2019). A note on generalized Leonardo numbers. Notes on Number Theory and Discrete Mathematics, 25(3), 97–101.
11. Shannon, A. G., & Deveci, O. (2022). A note on generalized and extended Leonardo sequences. Notes on Number Theory and Discrete Mathematics, 28(1), 109–114.
12. Shattuck, M. A., & Wagner, C. G. (2007). Some generalized Fibonacci polynomials. Journal of Integer Sequences, 10, Article 07.5.3.
13. Soykan, Y. (2021). Generalized Leonardo numbers. Journal of Progressive Research in Mathematics, 18(4), 58–84.
14. Wagner, C. G. (1996). Generalized Stirling and Lah numbers. Discrete Mathematics, 160, 199–218.
15. Werman, M., & Zeilberger, D. (1986). A bijective proof of Cassini’s Fibonacci identity. Discrete Mathematics, 58, 109.

Manuscript history

• Received: 19 July 2022
• Revised: 23 November 2022
• Accepted: 1 December 2022
• Online First: 5 December 2022