Gaussian Behavior and Geometric Gaps in Decompositions from Recurrences with Zero Coefficients

TL;DR

This paper explores the statistical properties of number decompositions in zero linear recurrence relations, revealing Gaussian summand distributions and geometric gap decay despite non-uniqueness.

Contribution

It demonstrates that key statistical behaviors like Gaussian summand counts and geometric gaps persist in ZLRRs, extending prior results beyond positive recurrence coefficients.

Findings

Number of summands converges to a Gaussian distribution.

Gaps between indices decay geometrically.

Legal decompositions grow exponentially at rate 2.

Abstract

Zeckendorf's theorem establishes a unique representation for positive integers as sums of non-consecutive Fibonacci numbers. This result has been generalized to Positive Linear Recurrence Sequences (PLRS), where key statistical properties, such as the Gaussian distribution of summands, depend on strictly positive recurrence coefficients. This paper investigates the consequences of relaxing this condition by studying \textit{Zero Linear Recurrence Relations (ZLRRs)}, where the leading coefficient is zero ($c_1=0$). Focusing on the \textit{Lagonacci sequence} ($Z_{n+1}=Z_{n-1}+Z_{n-2}$) as a primary case study, we demonstrate that while the uniqueness of decompositions is lost, fundamental statistical behaviors persist. We prove that the number of summands in the canonical greedy decomposition converges to a \textit{Gaussian distribution} and that the distribution of gaps between indices decays \textit{geometrically}. Furthermore, we utilize the \textit{principle of equivalence of ensembles} to show these properties are robust for a wide class of ZLRRs. Finally, we quantify the non-uniqueness of these systems, proving that the number of legal decompositions grows \textit{exponentially} at a rate $\alpha =2$, significantly exceeding the growth of the underlying sequence.