Relationships between Fibonacci-type sequences and Golden-type ratios

R. Patrick Vernon
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 24, 2018, Number 2, Pages 85–89
DOI: 10.7546/nntdm.2018.24.2.85-89
Full paper (PDF, 158 Kb)

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Authors and affiliations

R. Patrick Vernon
Department of Mathematics, Xavier University of Louisiana
1 Drexel Drive, New Orleans, LA, USA 70124

Abstract

The classical Fibonacci sequence is defined so that the first two terms are each equal to 1, and each term after this is the sum of the two terms immediately before it. The golden ratio is the ratio of the longer to shorter side of a rectangle with the property that if we remove a square from the rectangle such that the remainder is also a rectangle, that the old and new rectangles are proportional. Johannes Kepler showed that if we take the sequence of ratios of consecutive Fibonacci numbers, the limit of this sequence is the golden ratio [5]. In this paper, we give a higher dimension extension of Fibonacci sequences and golden ratios and provide a connection between the two.

Keywords

• Fibonacci sequence
• Golden ratio

2010 Mathematics Subject Classification

• 11B39

References

1.  Krcadinac, V. (2006) A new generalization of the golden ratio, Fibonacci Quarterly, 44 (4), 335–340.
2. Marohnic, L., & Strmecki, T. (2012) Plastic Number: Construction and Applications, Advanced Research in Scientific Areas, 1 (1), 1523–1528.
3. Stewart, I. (1996) Tales of a Neglected Number, Scientific American, 274 (6), 93.
4. Szczyrba, I. (2016) On the Existence of Ratio Limits of Weighted n-generalized Fibonacci Sequences with Arbitrary Initial Conditions, arXiv:1604.02361 [math.NT].
5. Tattersall, J. J. (2005) Elementary number theory in nine chapters. (2nd ed.) Cambridge University Press. p. 28.