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

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 $\mathit{\text{Zero}}\mathit{\text{Linear}}\mathit{\text{Recurrence}}\mathit{\text{Relations}}\mathit{\text{(ZLRRs)}}$, where the leading coefficient is zero ($c_1=0$). Focusing on the $\mathit{\text{Lagonacci}}\mathit{\text{sequence}}$ ($Z_{n+1}=Z_{n-1}+Z_{n-2}$) 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 $\mathit{\text{Gaussian}}\mathit{\text{distribution}}$ and that the distribution of gaps between indices decays $\mathit{\text{geometrically}}$. Furthermore, we utilize the $\mathit{\text{principle}}\mathit{\text{of}}\mathit{\text{equivalence}}\mathit{\text{of}}\mathit{\text{ensembles}}$ 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 $\mathit{\text{exponentially}}$ at a rate $\alpha =2$, significantly exceeding the growth of the underlying sequence.