Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-11T08:54:04.578Z Has data issue: false hasContentIssue false
  • English
  • Français

From golden-ratio equalities to Fibonacci and Lucas identities

Published online by Cambridge University Press:  23 January 2015

Martin Griffiths
Martin Griffiths*
Affiliation:
Mathematical Institute, University of OxfordOX1 3LB
Get access Rights & Permissions [Opens in a new window]

Extract

We demonstrate here a remarkably simple method for deriving a large number of identities involving the Fibonacci numbers, Lucas numbers and binomial coefficients. As will be shown, this is based on the utilisation of some straightforward properties of the golden ratio in conjunction with a result concerning irrational numbers. Indeed, for the simpler cases at least, the derivations could be understood by able high-school students. In particular, we avoid the use of exponential generating functions, matrix methods, Binet's formula, involved combinatorial arguments or lengthy algebraic manipulations.

Type
Articles
Information
The Mathematical Gazette , Volume 97 , Issue 539 , July 2013 , pp. 234 - 241
Copyright
Copyright © The Mathematical Association 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Knott, R., Fibonacci and golden ratio formulas (2012). http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibFormulae.html Google Scholar
2. Knuth, D. E., The art of computer programming, Volume I, Addison-Wesley (1968).Google Scholar
3. Benjamin, A. T., Eustis, A. K., and Plott, S. S., The 99th Fibonacci identity, The Electron. J. Combin. 15 (2008) R34.Google Scholar
4. Benjamin, A. T. and Quinn, J., Proofs that really count: the art of combinatorial proof, Mathematical Association of America (2003).Google Scholar
5. Carlitz, L. and Ferns, H. H., Some Fibonacci and Lucas identities, Fibonacci Quart. 8 (1970) pp. 61–73.Google Scholar
6. Layman, J. W., Certain general binomial-Fibonacci sums, Fibonacci Quart. 15 (1977) pp. 362–366.Google Scholar
7. Rabinowitz, S., Algorithmic manipulation of Fibonacci identities, in Bergum, G. E., Philippou, A. N., and Horodam, A. F. (Editors), Applications of Fibonacci Numbers: Proceedings of the Sixth International Research Conference on Fibonacci numbers and their applications, Kluwer Academic (1996).Google Scholar
8. Burton, D., Elementary number theory, McGraw-Hill (1998).Google Scholar
9. Hardy, G. H. and Wright, E. M., An introduction to the theory of numbers (6th edn), Oxford University Press (2008).Google Scholar
10. Church, C. A. and Bicknell, M., Exponential generating functions for Fibonacci identities, Fibonacci Quart. 11 (1973) pp. 275–281.Google Scholar
11. Griffiths, M., Families of sequences from a class of multinomial sums, J. Integer sequences 15 (2012) Article 12.1.8.Google Scholar
12. Sloane, N. J. A., The on-line encyclopedia of integer sequences (2012). https://oeis.org/ Google Scholar