Gaussian Behavior in Generalized Zeckendorf Decompositions

A beautiful theorem of Zeckendorf states that every integer can be written uniquely as a sum of nonconsecutive Fibonacci numbers \(\{F_{n}\}_{n=1}^{\infty }\); Lekkerkerker proved that the average number of summands for integers in [F n , F n+1) is \(n/(\varphi ^{2} + 1)\), with φ 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 → ∞, and comment on generalizations to related decompositions. For example, every integer can be written uniquely as a sum of the ± 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 \(-(21 - 2\varphi )/(29 + 2\varphi ) \approx -0.551058\).

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