Gaussian Behavior in Generalized Zeckendorf Decompositions

TL;DR

This paper proves that the distribution of the number of summands in Zeckendorf decompositions converges to a Gaussian distribution and explores related decompositions with signed Fibonacci sums, revealing normal and bivariate normal behaviors.

Contribution

It establishes the Gaussian limit law for the number of summands in Zeckendorf decompositions and extends the analysis to signed decompositions with explicit correlation.

Findings

Distribution of summands converges to a Gaussian as n increases.

Signed decompositions' positive and negative summands follow a bivariate normal distribution.

Correlation between positive and negative summands is approximately -0.551.

Abstract

A beautiful theorem of Zeckendorf states that every integer can be written uniquely as a sum of non-consecutive Fibonacci numbers $\{F_n\}_{n=1}^{\infty}$; Lekkerkerker proved that the average number of summands for integers in $[F_n, F_{n+1})$ is $n/(\phi^2 + 1)$, with $\phi$ the golden mean. Interestingly, the higher moments seem to have been ignored. We discuss the proof that the distribution of the number of summands converges to a Gaussian as $n \to \infty$, and comment on generalizations to related decompositions. For example, every integer can be written uniquely as a sum of the $\pm F_n$'s, such that every two terms of the same (opposite) sign differ in index by at least 4 (3). The distribution of the numbers of positive and negative summands converges to a bivariate normal with computable, negative correlation, namely $-(21-2\phi)/(29+2\phi) \approx -0.551058$.