1. Summand minimality and asymptotic convergence of generalized Zeckendorf decompositions.
- Author
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Cordwell, Katherine, Hlavacek, Max, Huynh, Chi, Miller, Steven J., Peterson, Carsten, and Vu, Yen Nhi Truong
- Subjects
STOCHASTIC convergence ,DECOMPOSITION method ,RECURSIVE sequences (Mathematics) ,MAXIMA & minima ,POLYNOMIALS ,INTEGERS ,NUMBER theory ,NUMBER systems - Abstract
Given a recurrence sequence H, with Hn=c1Hn-1+⋯+ctHn-t where ci∈N0 for all i and c1,ct≥1, the generalized Zeckendorf decomposition (gzd) of m∈N0 is the unique representation of m using H composed of blocks lexicographically less than σ=(c1,⋯,ct). We prove that the gzd of m uses the fewest number of summands among all representations of m using H, for all m, if and only if σ is weakly decreasing. We develop an algorithm for moving from any representation of m to the gzd, the analysis of which proves that σ weakly decreasing implies summand minimality. We prove that the gzds of numbers of the form v0Hn+⋯+vℓHn-ℓ converge in a suitable sense as n→∞; furthermore we classify three distinct behaviors for this convergence. We use this result, together with the irreducibility of certain families of polynomials, to exhibit a representation with fewer summands than the gzd if σ is not weakly decreasing. [ABSTRACT FROM AUTHOR]
- Published
- 2018
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