Summand minimality and asymptotic convergence of generalized Zeckendorf decompositions View Full Text


Ontology type: schema:ScholarlyArticle      Open Access: True


Article Info

DATE

2018-12

AUTHORS

Katherine Cordwell, Max Hlavacek, Chi Huynh, Steven J. Miller, Carsten Peterson, Yen Nhi Truong Vu

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. More... »

PAGES

43

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s40993-018-0137-7

DOI

http://dx.doi.org/10.1007/s40993-018-0137-7

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1107500267


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