The Golden Ratio family and the Binet equation

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_n(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 The Fibonacci Quarterly, by Horadam and Subba Rao in theBulletin 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

1. Cook, C. K., & Shannon, A.G. (2006) Generalized Fibonacci and Lucas Sequences with Pascal-type Arrays. Notes on Number Theory & Discrete Mathematics. 12(4): 1–9.
2. Filipponi, P. (1991) A Note on a Class of Lucas Sequences. The Fibonacci Quarterly.29(3), 256–263.
3. Horadam, A. F. (1966) Generalizations of Two Theorems of K. Subba Rao. Bulletin of the Calcutta Mathematical Society. 58, 23–29.
4. Jarden, D. (1966) Recurring Sequences. Jerusalem: Riveon Lematematika.
5. Kapur, J. N. (1976) Generalized Pascal’s Triangles and Generalized Fibonacci Numbers. Pure and Applied Mathematical Sciences. 3(1–2), 93–100.
6. Knott, R. (2015) Fibonacci Numbers and the Golden Section. Available online: http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fib.html.
7. Leyendekkers, J. V., Shannon, A. G. (2013) The Pascal–Fibonacci Numbers. Notes on Number Theory and Discrete Mathematics. 19(3), 5–11.
8. Leyendekkers, J. V., Shannon, A. G. (2015) The Golden Ratio Family and Generalized Fibonacci Numbers. Journal of Advances in Mathematics. 10(1), 3130–3137.
9. Leyendekkers, J. V., Shannon, A. G., Rybak, J. M. (2007) Pattern Recognition: Modular Rings and Integer Structure. North Sydney: Raffles KvB Monograph No.9.
10. Livio, Mario. 2002. The Golden Ratio. New York: Golden Books.
11. Lucas, E. (1969) The Theory of Simply Periodic Numerical Functions. (Edited by D.A.Lind; translated by S. Kravitz.) San Jose: The Fibonacci Association.
12. Monzingo, M. G. (1980) An Observation Concerning ‘Binet’s Formula Generalized’. In V.E. Hoggatt Jr, M. Bicknell-Johnson (eds). A Collection of Manuscripts related to the Fibonacci Sequence. Santa Clara: The Fibonacci Association, 93–94.
13. Peelle, H. A. (1975) Euclid, Fibonacci and Pascal – Recursed! International Journal of Mathematical Education in Science and Technology. 6(4), 395–405.
14. Rota, G.-C., Doubilet, P., Greene, C., Kahaner, D., Odlyzko, A., Stanley, R. (1975) Finite Operator Calculus. London: Academic Press, Ch.2.
15. Shannon, A. G. (1975) Fibonacci analogs of the classical polynomials. Mathematics Magazine. 48(3), 3–7.
16. Shannon, A. G. (1983) Intersections of second order linear recursive sequences. The Fibonacci Quarterly. 21(1), 6–12.
17. 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 & Technology. 37 (7), 825–831.
18. Shannon, A. G., Wong, C.K. (2010) Some Properties of Generalized Third Order Pell Numbers. Congressus Numerantium. 201, 345–354.
19. Shannon, A. G., Cook, C. K., Hillman, R. A. (2015) An Extension of Some Results due to Jarden. Journal of Advances in Mathematics. 9(8), 2937–2945.
20. Shannon, A. G., Turner, J. C., Atanassov, K.T. (1991) A Generalized Tableau Associated with Colored Convolution Trees. Discrete Mathematics. 92, 329–340.
21. Sloane, N.J.A., & Plouffe, S. (1995) The Encyclopedia of Integer Sequences. San Diego,CA: Academic Press; + Sloane, N.J.A. 1964+. https://oeis.org/.
22. Subba Rao, K. (1954) Some Properties of Fibonacci Numbers – I. Bulletin of the Calcutta Mathematical Society. 46, 253–257.
23. Subba Rao, K. (1959) Some Properties of Fibonacci Numbers – II. Mathematics Student.27, 19–23.
24. Whitford, A.K. (1977) Binet’s Formula Generalized. The Fibonacci Quarterly. 15(1), 21,24, 29.