Generalizing Zeckendorf's Theorem to f -decompositions

Demontigny, Philippe; Do, Thao; Kulkarni, Archit; Miller, Steven J.; Moon, David; Varma, Umang

doi:10.1016/j.jnt.2014.01.028

Cited by 32 publications

(42 citation statements)

References 12 publications

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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

Combinatorial and Additive Number Theory III

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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

Combinatorial and Additive Number Theory III

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“…By Theorem 1.5 in [6], since f (j) is periodic, we know that there is a single recurrence relation for our sequence. The proof of Theorem 1.5 in [6] gives us an algorithm for computing the single recurrence relation.…”

Section: Recurrence Relationsmentioning

confidence: 85%

“…Through an immediate application of Theorems 1.2 and 1.3 from [6] we can establish that for any n, m ∈ N, (n, m)-bin legal decompositions are unique and we get Proposition 1.1. Proposition 1.1.…”

Section: Introductionmentioning

confidence: 87%

“…In 1972 Edouard Zeckendorf proved that any positive integer can be uniquely decomposed as a sum of non-consecutive Fibonacci numbers provided we use the recurrence F 1 = 1, F 2 = 2, and F n = F n−1 + F n−2 for n ≥ 3 [13]. Since then numerous researchers have generalized Zeckendorf's theorem to other recurrence relations [1,6,7,9,11,12]. Most work involved recurrence relations with positive leading terms, called positive linear recurrences.…”

Section: Introductionmentioning

confidence: 99%

“…As described in [6], this notion of legal decompositions is an f -decomposition defined by the function f :…”

Section: Introductionmentioning

confidence: 99%

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Bin decompositions

Gotshall

Harris

Nelson

et al. 2019

Involve

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It is well known that every positive integer can be expressed as a sum of nonconsecutive Fibonacci numbers provided the Fibonacci numbers satisfy Fn = Fn−1 +Fn−2 for n ≥ 3, F1 = 1 and F2 = 2. In this paper, for any n, m ∈ N we create a sequence called the (n, m)-bin sequence with which we can define a notion of a legal decomposition for every positive integer. These sequences are not always positive linear recurrences, which have been studied in the literature, yet we prove, that like positive linear recurrences, these decompositions exist and are unique. Moreover, our main result proves that the distribution of the number of summands used in the (n, m)-bin legal decompositions displays Gaussian behavior. 2010 Mathematics Subject Classification. 11B39, 65Q30, 60B10. Key words and phrases. Zeckendorf decompositions, bin decompositions, Gaussian behavior, integer decompositions. P. E. Harris was supported by NSF award DMS-1620202. arXiv:1807.03918v1 [math.CO]

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Benford Behavior of Generalized Zeckendorf Decompositions

Best

Dynes

Edelsbrunner

et al. 2017

Springer Proceedings in Mathematics &Amp; Statistics

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We prove connections between Zeckendorf decompositions and Benford's law. Recall that if we define the Fibonacci numbers by F 1 = 1, F 2 = 2 and F n+1 = F n + F n−1 , every positive integer can be written uniquely as a sum of non-adjacent elements of this sequence; this is called the Zeckendorf decomposition, and similar unique decompositions exist for sequences arising from recurrence relations of the form G n+1 = c 1 G n + · · · + c L G n+1−L with c i positive and some other restrictions. Additionally, a set S ⊂ Z is said to satisfy Benford's law base 10 if the density of the elements in S with leading digit d is log 10 (1 + 1 d ); in other words, smaller leading digits are more likely to occur. We prove that as n → ∞ for a randomly selected integer m in [0, G n+1 ) the distribution of the leading digits of the summands in its generalized Zeckendorf decomposition converges to Benford's law almost surely. Our results hold more generally: one obtains similar theorems to those regarding the distribution of leading digits when considering how often values in sets with density are attained in the summands in the decompositions.

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Generalizing Zeckendorf's Theorem to f -decompositions

Cited by 32 publications

References 12 publications

The Zeckendorf Game

The Zeckendorf Game

Bin decompositions

Benford Behavior of Generalized Zeckendorf Decompositions

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