Generalized golden ratios and associated Pell sequences

A. G. Shannon and J. V. Leyendekkers
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 24, 2018, Number 3, Pages 103–110
DOI: 10.7546/nntdm.2018.24.3.103-110
Full paper (PDF, 105 Kb)

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Authors and affiliations

A. G. Shannon
Warrane College, The University of New South Wales
NSW 2033, Australia

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

Abstract

This paper considers generalizations of the golden ratio based on an extension of the Pell recurrence relation. These include related partial difference equations. It develops generalized Pell and Companion-Pell numbers and shows how they can yield elegant generalizations of Fibonacci and Lucas identities. This sheds light on the format of the original identities, such as the Simson formula, to distinguish what is significant and substantial from what is incidental or accidental.

Keywords

• Golden ratio
• Fibonacci numbers
• Lucas numbers
• Pell numbers
• Companion-Pell numbers
• Simson’s identity
• Binet formula
• Recurrence relations
• Difference equations
• Pythagorean triples

2010 Mathematics Subject Classification

• 11B39

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