2011/07/14 by Miller, Steven J., Wang, Yinghui · 1 citation
#11B39 (primary) 65Q30 #60B10 (secondary) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1107.2718
A beautiful theorem of Zeckendorf states that every integer can be written uniquely as a sum of non-consecutive Fibonacci numbers \Fn\n=1∞; Lekkerkerker proved that the average number of summands for integers in [Fn, Fn+1) is n/(ϕ2 + 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 ± Fn'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 -(21-2ϕ)/(29+2ϕ) ≈ -0.551058.