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Some
combinatorics from Zeckendorf representations
Tyler Ball, Rachel Chaiser, Dean Dustin, Tom Edgar and Paul Lagarde
Vol. 12 (2019), No. 7, 1241–1260
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Abstract |
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We explore some properties of the so-called Zeckendorf representations of integers, where we write an integer as a sum of distinct, nonconsecutive Fibonacci numbers. We examine the combinatorics arising from the arithmetic of these representations, with a particular emphasis on understanding the Zeckendorf tree that encodes them. We introduce some possibly new results related to the tree, allowing us to develop a partial analog to Kummer’s classical theorem about counting the number of “carries” involved in arithmetic. Finally, we finish with some conjectures and possible future projects related to the combinatorics of these representations. |
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Keywords
Fibonacci, Zeckendorf, digital dominance order
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Mathematical Subject Classification 2010
Primary: 06A07, 11B39, 11B75, 11Y55
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Milestones
Received: 28 March 2019
Accepted: 10 June 2019
Published: 12 October 2019
Communicated by Arthur T. Benjamin
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Authors |
| Tyler Ball | |
| Clover Park High School Lakewood, WA United States |
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| Rachel Chaiser | |
| University of Colorado Boulder Boulder, CO United States |
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| Dean Dustin | |
| University of Nebraska Lincoln, NE United States |
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| Tom Edgar | |
| Department of Mathematics Pacific Lutheran University Tacoma, WA United States |
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| Paul Lagarde | |
| South Merrimack Christian
Academy Merrimack, NH United States |
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