Generalizing Zeckendorf's Theorem: The Kentucky Sequence

TL;DR

This paper extends Zeckendorf's theorem to a broader class of sequences, demonstrating unique legal decompositions, Gaussian distribution of summands, and geometric gap decay.

Contribution

It introduces a new sequence generalization with a zero initial term, maintaining key properties like uniqueness and probabilistic behaviors.

Findings

Unique legal decompositions for the generalized sequence

Gaussian distribution of the number of summands

Geometric decay in gap distribution

Abstract

By Zeckendorf's theorem, an equivalent definition of the Fibonacci sequence (appropriately normalized) is that it is the unique sequence of increasing integers such that every positive number can be written uniquely as a sum of non-adjacent elements; this is called a legal decomposition. Previous work examined the distribution of the number of summands and the spacings between them, in legal decompositions arising from the Fibonacci numbers and other linear recurrence relations with non-negative integral coefficients. Many of these results were restricted to the case where the first term in the defining recurrence was positive. We study a generalization of the Fibonacci numbers with a simple notion of legality which leads to a recurrence where the first term vanishes. We again have unique legal decompositions, Gaussian behavior in the number of summands, and geometric decay in the distribution of gaps.