An extremal problem related to the Fibonacci sequence

K. T. Atanassov, R. D. Knott, R. L. Ollerton and A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 12, 2006, Number 2, Pages 13—20
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Authors and affiliations

K. T. Atanassov
Centre for Biomedical Engineering, Bulgarian Academy of Sciences,
Sofia-1113, Bulgaria

R. D. Knott
92 Pennine Road, Horwich,
Bolton, BL6 7HW, United Kingdom

R. L. Ollerton
University of Western Sydney, Penrith Campus DC1797, Australia

A. G. Shannon
Warrane College, The University of New South Wales, 1465 &
KvB Institute of Technology, North Sydney, NSW, 2060, Australia

Abstract

This paper continues our study of Fibonacci inequalities. For the set An = {Fn−1, 4Fn−2, …, (n−2)2F2} with kth element given by ak = k2Fn−k, it is proved that the unique maximal element is given by a* = a4 = 16Fn−4, n ≥ 9.

AMS Classification

  • 11B39

References

  1. K.T. Atanassov, Ron Knott, Kiyota Ozeki, A.G. Shannon, László Szalay. “Inequalities Among Related Pairs of Fibonacci Numbers.” The Fibonacci Quarterly. 41 (2003):20-22.
  2. S. Vajda, Fibonacci and Lucas Numbers and the Golden Section: Theory and Applications. Chichester: Ellis Horwood, 1989.
  3. N. Gauthier, “Two Fibonacci Sums – a Variation.” Mathematical Gazette. 81 (1997): 85-88.
  4. P. Glaister. “Fibonacci Sums of the Type .” Mathematical Gazette. 79 (1995): 364-367.
  5. A.F. Horadam. “Basic Properties of a Certain Generalized Sequence of Numbers.” The Fibonacci Quarterly. 3 (1965): 161-176.
  6. Wolfram Research Inc http://mathworld.wolfram.com/FibonacciNumber.html.
  7. Kiyota Ozeki, “On Weighted Fibonacci and Lucas Sums.” The Fibonacci Quarterly, 43 (2005): 104-107.

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Cite this paper

APA

Atanassov, K. T., Knot, R. D., Ollerton, R. L., & Shannon, A. G. (2006). An extremal problem related to the Fibonacci sequence. Notes on Number Theory and Discrete Mathematics, 12(2), 13-20.

Chicago

Atanassov, K. T., R. D. Knott, R. L. Ollerton, and A. G. Shannon. “An Extremal Problem Related to the Fibonacci Sequence.” Notes on Number Theory and Discrete Mathematics 12, no. 2 (2006): 13-20.

MLA

Atanassov, K. T., R. D. Knott, R. L. Ollerton, and A. G. Shannon. “An Extremal Problem Related to the Fibonacci Sequence.” Notes on Number Theory and Discrete Mathematics 12.2 (2006): 13-20. Print.

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