Some characteristics of the Golden Ratio family

J. V. Leyendekkers and A. G. Shannon
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 22, 2016, Number 3, Pages 84—89
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Authors and affiliations

J. V. Leyendekkers
Faculty of Science, The University of Sydney, NSW 2006, Australia

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

Abstract

Research into properties of generalizations of the Golden Ratio has considered various forms of extreme and mean ratios. This is considered here too within the framework of a family of surds, ½ √(1+a), and generalized Fibonacci numbers, Fn(a), with the ordinary Fibonacci numbers being the particular case when a = 5.

Keywords

  • Modular rings
  • Golden Ratio
  • Infinite series
  • Binet formula
  • Right-end-digits
  • Fibonacci sequence

AMS Classification

  • 11B39
  • 11B50

References

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  2. Coxeter, H.S.M. (1953) The Golden Section, Phyllotaxis, and Wythoff’s Game. Scripta Mathematica. 19, 135–143.
  3. Crăciun, I., Invan, D., & Popa, D. (2015) Generalized Golden Ratios defined by means. Applied Mathematics and Computation. 250(1), 221–227.
  4. Kapur, J. N. (1988) Some generalizations of the golden ratio. International Journal of Mathematical Education in Science and Technology. 19(4), 511–517.
  5. Leyendekkers, J. V., & Shannon, A.G. (2015) The Golden Ratio Family and Generalized Fibonacci Numbers. Journal of Advances in Mathematics. 10(1), 3130–3137.
  6. Leyendekkers, J. V., & Shannon, A.G. (2015) The Sum of Squares for Primes. Notes on Number Theory and Discrete Mathematics. 21(4), 17–21.
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  8. Livio, Mario. 2002. The Golden Ratio. New York: Golden Books.
  9. Shannon, A. G., & Bernstein, L. (1973) The Jacobi–Perron Algorithm and the Algebra of Recursive Sequences. Bulletin of the Australian Mathematical Society. 8(4), 261–277.
  10. Zaremba, S. K. (1970) A Remarkable Lattice Generated by Fibonacci Numbers. The Fibonacci Quarterly. 8(2), 185–198.

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Cite this paper

APA

Leyendekkers, J. V., & Shannon, A. G. (2016). Some characteristics of the Golden Ratio family, Notes on Number Theory and Discrete Mathematics, 22(3), 84-89.

Chicago

Leyendekkers, J. V. and A. G. Shannon “Some Characteristics of the Golden Ratio Family.” Notes on Number Theory and Discrete Mathematics 22, no. 3 (2016): 84-89.

MLA

Leyendekkers, J. V. and A. G. Shannon, “Some Characteristics of the Golden Ratio Family.” Notes on Number Theory and Discrete Mathematics 22.3 (2016): 84-89. Print.

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