From Fibonacci numbers to central limit type theorems
- Miller, Steven J.
- Wang, Yinghui
DOIAbstract
AbstractA beautiful theorem of Zeckendorf states that every integer can be written uniquely as a sum of non-consecutive Fibonacci numbers {Fn}n=1∞. Lekkerkerker (1951–1952) [13] proved the average number of summands for integers in [Fn,Fn+1) is n/(φ2+1), with φ the golden mean. This has been generalized: given non-negative integers c1,c2,…,cL with c1,cL>0 and recursive sequence {Hn}n=1∞ with H1=1, Hn+1=c1Hn+c2Hn−1+⋯+cnH1+1 (1⩽n<L) and Hn+1=c1Hn+c2Hn−1+⋯+cLHn+1−L (n⩾L), every positive integer can be written uniquely as ∑aiHi under natural constraints on the aiʼs, the mean and variance of the numbers of summands for integers in [Hn,Hn+1) are of size n, and as n→∞ the distribution of the number of summands converges to a Gaussian. Previous app...
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