Corrigendum to “generalized Zeckendorf expansions”

Filipponi, Piero; Grabner, Peter J.; Nemes, István; Pethö, Attila; Tichy, Robert F.

doi:10.1016/0893-9659(94)90087-6

Cited by 29 publications

(17 citation statements)

References 1 publication

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“…for uniqueness. The Zeckendorf theorem has been generalized many times (see for example [Br,CFHMN1,CFHMN2,CFHMNPX,Day,DDKMMV,DFFHMPP,FGNPT,Fr,GTNP,Ha,Ho,Ke,LT,Len,Ste1,Ste2]); we follow the terminology used by Miller and Wang [MW1].…”

Section: Introductionmentioning

confidence: 99%

The Zeckendorf Game

Baird-Smith

Epstein

Flint

et al. 2019

Springer Proceedings in Mathematics &Amp; Statistics

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Zeckendorf proved that every positive integer n can be written uniquely as the sum of non-adjacent Fibonacci numbers; a similar result, though with a different notion of a legal decomposition, holds for many other sequences. We use these decompositions to construct a two-player game, which can be completely analyzed for linear recurrence relations of the form G n = k i=1 cG n−i for a fixed positive integer c (c = k − 1 = 1 gives the Fibonaccis). Given a fixed integer n and an initial decomposition of n = nG 1 , the two players alternate by using moves related to the recurrence relation, and whomever moves last wins. The game always terminates in the Zeckendorf decomposition, though depending on the choice of moves the length of the game and the winner can vary. We find upper and lower bounds on the number of moves possible; for the Fibonacci game the upper bound is on the order of n log n, and for other games we obtain a bound growing linearly with n. For the Fibonacci game, Player 2 has the winning strategy for all n > 2. If Player 2 makes a mistake on his first move, however, Player 1 has the winning strategy instead. Interestingly, the proof of both of these claims is non-constructive.

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Section: Introductionmentioning

confidence: 99%

The Zeckendorf Game

Baird-Smith

Epstein

Flint

et al. 2019

Springer Proceedings in Mathematics &Amp; Statistics

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“…For every such recurrence relation there is a notion of "legal decomposition" with which all positive integers have a unique decomposition as a non-negative integer linear combination of terms from the sequence, and the distribution of the number of summands of integers in [G n , G n+1 ) converges to a Gaussian. There is an extensive literature for this subject; see [Al,BCCSW,Day,GT,Ha,Ho,Ke,Len,MW1,MW2] for results on uniqueness of decomposition, [DG,FGNPT,GTNP,KKMW,Lek,LamTh,MW1,St] for Gaussian behavior, and [BBGILMT] for recent work on the distribution of gaps between summands. An alternative definition of the Fibonacci sequence can be framed in terms of the Zeckendorf non-consecutive condition: The Fibonacci sequence (beginning F 1 = 1, F 2 = 2) is the unique increasing sequence of natural numbers such that every positive integer can be written uniquely as a sum of non-consecutive terms from the sequence.…”

Section: Introductionmentioning

confidence: 99%

Generalizing Zeckendorf's Theorem to f -decompositions

Demontigny

Kulkarni

et al. 2014

Journal of Number Theory

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“…Below we report on some of the previous work on properties of Generalized Zeckendorf decompositions for certain sequences, and then discuss our new generalizations to two-dimensional sequences. There is now an extensive literature on the subject (see for example [BDEMMTTW,BILMT,Br,CFHMN1,Day,DDKMMV,FGNPT,Fr,GTNP,Ha,Ho,HW,Ke,MW1,MW2,Ste1,Ste2] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Gaussian Behavior in Zeckendorf Decompositions From Lattices

Chen¹,

Chen²,

Guo³

et al. 2018

Preprint

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Zeckendorf's Theorem states that any positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers. We consider higher-dimensional lattice analogues, where a legal decomposition of a number n is a collection of lattice points such that each point is included at most once. Once a point is chosen, all future points must have strictly smaller coordinates, and the pairwise sum of the values of the points chosen equals n. We prove that the distribution of the number of summands in these lattice decompositions converges to a Gaussian distribution in any number of dimensions. As an immediate corollary we obtain a new proof for the asymptotic number of certain lattice paths.

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Corrigendum to “generalized Zeckendorf expansions”

Cited by 29 publications

References 1 publication

The Zeckendorf Game

The Zeckendorf Game

Generalizing Zeckendorf's Theorem to f -decompositions

Gaussian Behavior in Zeckendorf Decompositions From Lattices

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