From Fibonacci numbers to central limit type theorems

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 (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