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arXiv · 1810.03053

Limiting Distributions in Generalized Zeckendorf Decompositions

Alexandre Gueganic·Granger Carty·Yujin H. Kim·Steven J. Miller·Alina Shubina·Shannon Sweitzer·Eric Winsor·Jianing Yang
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Abstract

An equivalent definition of the Fibonacci numbers is that they are the unique sequence such that every integer can be written uniquely as a sum of non-adjacent terms. We can view this as we have bins of length 1, we can take at most one element from a bin, and if we choose an element from a bin we cannot take one from a neighboring bin. We generalize to allowing bins of varying length and restrictions as to how many elements may be used in a decomposition. We derive conditions on when the resulting sequences have uniqueness of decomposition, and (similar to the Fibonacci case) when the number of summands converges to a Gaussian; the main tool in the proofs here is the Lyaponuv Central Limit Theorem.

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Alexandre Gueganic, Granger Carty, Yujin H. Kim, Steven J. Miller, Alina Shubina, Shannon Sweitzer, Eric Winsor, Jianing Yang. 2018-10-06. Limiting Distributions in Generalized Zeckendorf Decompositions. https://arxiv.org/abs/1810.03053

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