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

**Full paper (PDF, 105 Kb)**

## 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

- Livio, M. (2002) The Golden Ratio. Golden Books, New York.
- Atanassov, K., Atanassova, V., Shannon, A., & Turner, J. (2002) New Visual Perspectives on the Fibonacci Numbers. World Scientific, New York.
- Hoggatt, V. E., Jr. (1969) Fibonacci and Lucas Numbers. Houghton-Mifflin, Boston.
- Tee, G. J. (2003) Russian Peasant Multiplication and Egyptian Division in Zeckendorf Arithmetic. Australian Mathematical Society Gazette, 30, 267–276.
- Stakhov, O. (1997) Computer Arithmetic based on Fibonacci Numbers and Golden Section: New Information and Arithmetic Computer Foundations. Vinnitsa: Ukrainian Academy of Engineering Sciences, Ch.2.
- Graham, R. L., Knuth, D. E., & Patashnik, O. (1992) Concrete Mathematics: A Foundation for Computer Science. Addison-Wesley, Reading, MA.
- Shannon, A. G., & Leyendekkers, J. V. (2015) The Golden Ratio family and the Binet equation.
*Notes on Number Theory and Discrete Mathematics*, 21(2), 35–42. - Falcon, S. (2018) Some new formulas on the K-Fibonacci numbers. Journal of Advances in Mathematics, 14(1), 7439–7445.
- Horadam, A. F. (1965) Basic properties of a certain generalized sequence of numbers. The Fibonacci Quarterly, 3(3), 161–176.
- Cook, C. K., & Shannon, A. G. (2006) Generalized Fibonacci and Lucas sequences with Pascal-type arrays.
*Notes on Number Theory and Discrete Mathematics*, 12(4), 1–9. - Shannon, A. G., Anderson, P. G., & Horadam, A. F. (2006) Properties of Cordonnier, Perrin and van der Laan Numbers. International Journal of Mathematical Education in Science and Technology, 37(7), 825–831.
- Shannon, A. G., & Horadam, A. F. (2004) Generalized Pell numbers and polynomials. In: Fredric T. Howard (ed.) Applications of Fibonacci Numbers, Volume 9. Dordrecht/Boston/London: Kluwer, 213–224.
- Deveci, Ö., & Shannon, A. G. (2017) Pell–Padovan-circulant sequences and their applications.
*Notes on Number Theory and Discrete Mathematics*, 23(3), 100–114. - Shannon, A. G., & Leyendekkers, J. V. (2018) The Fibonacci Numbers and Integer Structure. Nova Science Publishers, New York.
- Leyendekkers, J. V., & Shannon, A. G. (2015) The Golden Ratio Family and generalized Fibonacci Numbers. Journal of Advances in Mathematics, 10(1), 3130–3137.
- Leyendekkers, J. V., & Rybak, J. M. (1995) Pellian sequences derived from Pythagorean triples. International Journal of Mathematical Education in Science and Technology, 26(6), 903–922.

## 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.