A. G. Shannon and J. V. Leyendekkers

Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132

Volume 21, 2015, Number 2, Pages 35–42

**Full paper (PDF, 101 Kb)**

## Details

### Authors and affiliations

A. G. Shannon

*Faculty of Engineering & IT, University of Technology
Sydney, NSW 2007, Australia &
Campion College
PO Box 3052, Toongabbie East, NSW 2146, Australia
*

J. V. Leyendekkers

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

### Abstract

The Golden Ratio can be considered as the first member of a family which can generate a set of generalized Fibonacci sequences. Here we consider some related problems in terms of the Binet form of these sequences, {*F _{n}*(

*a*)}, where the sequence of ordinary Fibonacci numbers can be expressed as {

*F*(5)} in this notation. A generalized Binet equation can predict all the elements of the Golden Ratio family of sequences. Identities analogous to those of the ordinary Fibonacci sequence are developed as extensions of work by Filipponi, Monzingo and Whitford in

_{n}*The Fibonacci Quarterly*, by Horadam and Subba Rao in the

*Bulletin of the Calcutta Mathematical Society*, within the framework of Sloane’s

*Online Encyclopedia of Interger Sequences*.

### Keywords

- Modular rings
- Golden ratio
- Infinite series
- Binet formula
- Fibonacci sequence
- Finite difference operators

### AMS Classification

- 11B39
- 11B50

### References

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## Related papers

- Spivey, R. J. (2019). Close encounters of the golden and silver ratios.
*Notes on Number Theory and Discrete Mathematics*, 25(3), 170-184. - Leyendekkers, J. V., & Shannon, A. G. (2016). Some Golden Ratio generalized Fibonacci and Lucas sequences.
*Notes on Number Theory and Discrete Mathematics*, 22(1), 33-41.

## Cite this paper

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.