Some combinatorics from Zeckendorf representations
Tyler E. Ball, Rachel Chaiser, Dean Dustin, Tom Edgar, Paul Lagarde · Involve a Journal of Mathematics · 2019
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.