J. V. Leyendekkers and A. G. Shannon

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

Volume 22, 2016, Number 1, Pages 33—41

**Download full paper: PDF, 206 Kb**

## Details

### Authors and affiliations

J. V. Leyendekkers

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

A. G. Shannon

*Emeritus Professor, University of Technology Sydney, NSW 2007, Australia*

Campion College, PO Box 3052, Toongabbie East, NSW 2146, Australia

Campion College, PO Box 3052, Toongabbie East, NSW 2146, Australia

### Abstract

Various characteristics of the ordinary Fibonacci and Lucas sequences, many known for centuries, are shown to be common to generalized sequences related to the Golden Ratio. Periodicity properties are also investigated.

### Keywords

- Unit digits (right-end-digits)
- Modular rings
- Golden Ratio
- reduced Pythagorean triples
- Fibonacci and Lucas numbers
- Pythagorean triples

### AMS Classification

- 11B39
- 11B50

### References

- Horadam, A. F. (1966) Generalizations of two theorems of K. Subba Rao. Bulletin of the Calcutta Mathematical Society, 58(1), 23–29.
- Kilic, E., & D. Tasci (2006) The generalized Binet formula, representation and sums of the generalized order-
*k*Pell numbers. Taiwanese Journal of Mathematics, 10(6), 1661–1670. - Leyendekkers, J. V., J. M. Rybak, & A. G. Shannon (1995) Integer class properties

associated with an integer matrix. Notes on Number Theory and Discrete Mathematics. 1(2), 53–59 - Leyendekkers, J. V., & A. G. Shannon (2015) The Golden Ratio Family and Generalized Fibonacci Numbers. Journal of Advances in Mathematics . 10(1), 3130–3137.
- Leyendekkers, J.V., A.G Shannon, J.M. Rybak. 2007. Pattern Recognition: Modular Rings and Integer Structure. North Sydney: Raffles KvB Monograph No.9.
- Leyendekkers, J.V., A.G Shannon. The sum of squares for primes. Notes on Number Theory and Discrete Mathematics. 21(4), 17–21.
- Livio, M. (2002)The Golden Ratio. New York: Golden Books.
- Shannon, A. G., A. F. Horadam, & S. N. Collings (1974) Some Fibonacci congruences. The Fibonacci Quarterly, 12(4), 351–354.
- Shannon, A. G., & A. F. Horadam (1994) Arrowhead curves in a tree of Pythagorean triples. International Journal of Mathematical Education in Science and Technology, 25(2), 255–261.
- Shannon, A. G., & J. V. Leyendekkers (2015) The Golden Ratio Family and the Binet equation. Notes on Number Theory and Discrete Mathematics, 21(2), 35–42.
- Subba Rao, K. (1959) Some properties of Fibonacci numbers – II. The Mathematics Student, 27(1), 19–23.
- Tee, G. J. (1991) Proof of Filipponi’s Conjecture on Non-Integer Sums of Fibonacci-Type Series. Gazette of the Australian Mathematical Society, 18, 161–164.
- Vajda, S. (1989) Fibonacci & Lucas Numbers, and the Golden Section: Theory and Applications. Chichester: Ellis Horwood.

## 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.
- Nagaraja, K. M., & Dhanya, P. (2020). Identities on generalized Fibonacci and Lucas numbers. Notes on Number Theory and Discrete Mathematics, 26 (3), 189-202.

## Cite this paper

APALeyendekkers, 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.

ChicagoLeyendekkers, J. V., and A. G. Shannon. “Some Golden Ratio generalized Fibonacci and Lucas sequences.” Notes on Number Theory and Discrete Mathematics 22, no. 1 (2016): 33-41.

MLALeyendekkers, J. V., and A. G. Shannon. “Some Golden Ratio generalized Fibonacci and Lucas sequences.” Notes on Number Theory and Discrete Mathematics 22.1 (2016): 33-41. Print.