**Anthony G. Shannon, Peter J.-S. Shiue, Shen C. Huang, Ali Balooch, Yu-Chung Liu**

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 30, 2024, Number 2, Pages 283–310

DOI: 10.7546/nntdm.2024.30.2.283-310

**Full paper (PDF, 319 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
*

Ali Balooch

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

Yu-Chung Liu

*School of Mathematical and Statistical Sciences, Clemson University
105 Sikes Hall, Clemson, SC 29634, USA
*

### Abstract

This paper extends known results of second and third order recursive sequences through extensive formulations of properties of the roots of their characteristic equations, some are old but most are new. They are applied to novel studies of , including their convergence criteria, and applied to many standard sequences, as particular cases of a generic . The detailed development of the algebra of the pertinent theorems, and their associated lemmas and corollaries, should open up new vistas for interested number theorists with the concluding results on series values.

### Keywords

- Arithmetic sequence
- Balancing sequence
- Fermat sequence
- Fibonacci sequence
- Geometric sequence
- Jacobsthal sequence
- Leonardo sequence
- Lucas sequence
- Mersenne sequence
- Padovan sequence
- Pell sequence
- Perrin sequence

### 2020 Mathematics Subject Classification

- 11B39
- 11A25

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### Manuscript history

- Received: 14 February 2023
- Revised: 14 May 2024
- Accepted: 18 May 2024
- Online First: 19 May 2024

### Copyright information

Ⓒ 2024 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

- 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

Shannon, A. G., Shiue, P. J.-S., Huang, S. C., Balooch, A., & Liu, Y.-C. (2024). Some infinite series summations involving linear recurrence relations of order 2 and 3. *Notes on Number Theory and Discrete Mathematics*, 30(2), 283-310, DOI: 10.7546/nntdm.2024.30.2.283-310.