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)**

## Details

### 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

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### Manuscript history

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

## Related papers

- Shannon, A. G. (2019). A note on generalized Leonardo numbers.
*Notes on Number Theory and Discrete Mathematics*, 25(3), 97–101. - 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. - Karataş, A. (2022). On complex Leonardo numbers.
*Notes on Number Theory and Discrete Mathematics*, 28(3), 458–465. - Shannon, A. G., Shiue, P. J.-S., & Huang, S. C. (2023). Notes on generalized and extended Leonardo numbers.
*Notes on Number Theory and Discrete Mathematics*, 29(4), 752-773.

## Cite this paper

Shattuck, M. (2022). Combinatorial proofs of identities for the generalized Leonardo numbers. *Notes on Number Theory and Discrete Mathematics*, 28(4), 778-790, DOI: 10.7546/nntdm.2022.28.4.778-790.