A. G. Shannon and J. V. Leyendekkers

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 24, 2018, Number 3, Pages 103—110

DOI: 10.7546/nntdm.2018.24.3.103-110

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## Details

### Authors and affiliations

A. G. Shannon

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

J. V. Leyendekkers

*Faculty of Science, The University of Sydney
NSW 2006, Australia
*

### Abstract

This paper considers generalizations of the golden ratio based on an extension of the Pell recurrence relation. These include related partial difference equations. It develops generalized Pell and Companion-Pell numbers and shows how they can yield elegant generalizations of Fibonacci and Lucas identities. This sheds light on the format of the original identities, such as the Simson formula, to distinguish what is significant and substantial from what is incidental or accidental.

### Keywords

- Golden ratio
- Fibonacci numbers
- Lucas numbers
- Pell numbers
- Companion-Pell numbers
- Simson’s identity
- Binet formula
- Recurrence relations
- Difference equations
- Pythagorean triples

### 2010 Mathematics Subject Classification

- 11B39

### References

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## Related papers

- Kosobutskyy, P. S. (2020). Phidias numbers as a basis for Fibonacci analogues. Notes on Number Theory and Discrete Mathematics, 26(1), 172-178.

## Cite this paper

Shannon, A. G., & Leyendekkers, J. V. (2018). Generalized golden ratios and associated Pell sequences. Notes on Number Theory and Discrete Mathematics, 24(3), 103-110, doi: 10.7546/nntdm.2018.24.3.103-110.