Generalizing Zeckendorf's Theorem to f-decompositions

Text. A beautiful theorem of Zeckendorf states that every positive integer can be uniquely decomposed as a sum of non-consecutive Fibonacci numbers {F-n}, where F-1 = 1, F-2 = 2 and Fn+1 = F-n + Fn-1. For general recurrences {G(n)} with nonnegative coefficients, there is a notion of a legal decomposition which again leads to a unique representation. We consider the converse question: given a notion of legal decomposition, construct a sequence {a(n)}such that every positive integer can be uniquely decomposed as a sum of a(n)'s. We prove this is possible for a notion of legal decomposition called f-decompositions. This notion generalizes existing notions such as base-b representations, Zeckendorf decompositions, and the factorial number system. Using this new perspective, we expand the range of Zeckendorf-type results, generalizing the scope of previous research. Finally, for specific classes of notions of decomposition we prove a Gaussianity result concerning the distribution of the number of summands in the decomposition of a randomly chosen integer. Video. For a video summary of this paper, please click here or visit http://youtu.be/hnYJwvOfzLo. (C) 2014 Elsevier Inc. All rights reserved.