**Anthony G. Shannon, Peter J.-S. Shiue and Shen C. Huang**

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 29, 2023, Number 4, Pages 752–773

DOI: 10.7546/nntdm.2023.29.4.752-773

**Full paper (PDF, 359 Kb)**

## Details

### Authors and affiliations

Anthony G. Shannon

*Warrane College, University of New South Wales
Kensington, NSW 2033, Australia*

Peter J.-S. Shiue

*Department of Mathematical Sciences, University of Nevada, Las Vegas
4505 S Maryland Pkwy, Las Vegas, NV 89154, USA*

Shen C. Huang

*Department of Mathematical Sciences, University of Nevada, Las Vegas
4505 S Maryland Pkwy, Las Vegas, NV 89154, USA*

### Abstract

This paper both extends and generalizes recently published properties which have been developed by many authors for elements of the Leonardo sequence in the context of second-order recursive sequences. It does this by considering the difference equation properties of the homogeneous Fibonacci sequence and the non-homogeneous properties of their Leonardo sequence counterparts. This produces a number of new identities associated with a generalized Leonardo sequence and its associated algorithm, as well as some combinatorial results which lead into elegant properties of hyper-Fibonacci numbers in contrast to their ordinary Fibonacci number analogues, and as a convolution of Fibonacci and Leonardo numbers.

### Keywords

- Binet formulas
- Leonardo sequences
- Generalized Leonardo sequence
- Extended Leonardo sequence
- Fibonacci sequences
- Hyper-Fibonacci sequences
- Recurrence relations
- Undetermined coefficients

### 2020 Mathematics Subject Classification

- 05A19
- 11B37
- 11B39

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### Manuscript history

- Received: 19 September 2023
- Accepted: 30 October 2023
- Online First: 27 November 2023

### Copyright information

Ⓒ 2023 by the Authors.

This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

## Related papers

- Alp, Y. (2023). Hybrid hyper-Fibonacci and hyper-Lucas numbers.
*Notes on Number Theory and Discrete Mathematics*, 29(1), 154–170. - 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, Ö. (2022). A note on generalized and extended Leonardo sequences.
*Notes on Number Theory and Discrete Mathematics*, 28(1), 109–114. - Shattuck, M. (2022). Combinatorial proofs of identities for the generalized Leonardo numbers.
*Notes on Number Theory and Discrete Mathematics*, 28(4), 778–790.

## Cite this paper

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, DOI: 10.7546/nntdm.2023.29.4.752-773.