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.
@article{arxiv.1810.03053,
title = {Limiting Distributions in Generalized Zeckendorf Decompositions},
author = {Alexandre Gueganic and Granger Carty and Yujin H. Kim and Steven J. Miller and Alina Shubina and Shannon Sweitzer and Eric Winsor and Jianing Yang},
journal= {arXiv preprint arXiv:1810.03053},
year = {2019}
}