A tiling approach to Fibonacci product identities

Artz, Jacob H.; Rowell, Michael

doi:10.2140/involve.2009.2.581

Cited by 4 publications

(11 citation statements)

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“…The Zeckendorf representation of 2 f m f n A Zeckendorf representation for 2 f m f n was given in [Filipponi and Hart 1998]. We provide a combinatorial proof for this identity, extending the combinatorial methods from [Artz and Rowell 2009].…”

Section: Woodmentioning

confidence: 89%

“…Gerdemann [2009] gave a combinatorial algorithm for finding the Zeckendorf representation of any particular m f n , but it does not give a general closed-form representation. Artz and Rowell [2009] found combinatorial proofs of certain Zeckendorf representations of f m f n originally proved in [Filipponi and Hart 1998] by other means:…”

Section: Preliminariesmentioning

confidence: 99%

“…In Section 4 we answer an open problem from [Artz and Rowell 2009] and present many new Fibonacci and Lucas product Zeckendorf representations.…”

Section: Woodmentioning

confidence: 99%

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2011

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Section: Woodmentioning

confidence: 89%

Section: Preliminariesmentioning

confidence: 99%

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2011

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“…In [Artz and Rowell 2009], the following theorem was given and an open problem was posed to find a combinatorial proof. The following proof gives an answer to the open question.…”

Section: Answering An Open Problem and New Zeckendorf Representationsmentioning

confidence: 99%

“…In Sections 2 and 3 we provide combinatorial proofs of additional Zeckendorf representations of Fibonacci and Lucas products given in [Filipponi and Hart 1998], namely those for 2 f m f n and L m L n . In Section 4 we answer an open problem from [Artz and Rowell 2009] and present many new Fibonacci and Lucas product Zeckendorf representations.…”

mentioning

confidence: 99%

Combinatorial proofs of Zeckendorf representations of Fibonacci and Lucas products

McGregor

Rowell

2011

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In 1998, Filipponi and Hart introduced many Zeckendorf representations of Fibonacci, Lucas and mixed products involving two variables. In 2008, Artz and Rowell proved the simplest of these identities, the Fibonacci product, using tilings. This paper extends the work done by Artz and Rowell to many of the remaining identities from Filipponi and Hart's work. We also answer an open problem raised by Artz and Rowell and present many Zeckendorf representations of mixed products involving three variables.

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Zeckendorf representation of multiplicative inverses modulo a Fibonacci number

2022

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Prempreesuk, Noppakaew, and Pongsriiam determined the Zeckendorf representation of the multiplicative inverse of 2 modulo $$F_n$$ F n , for every positive integer n not divisible by 3, where $$F_n$$ F n denotes the nth Fibonacci number. We determine the Zeckendorf representation of the multiplicative inverse of a modulo $$F_n$$ F n , for every fixed integer $$a \ge 3$$ a ≥ 3 and for all positive integers n with $$\gcd (a, F_n) = 1$$ gcd ( a , F n ) = 1 . Our proof makes use of the so-called base-$$\varphi $$ φ expansion of real numbers.

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A tiling approach to Fibonacci product identities

Cited by 4 publications

References 1 publication

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Combinatorial proofs of Zeckendorf representations of Fibonacci and Lucas products

Zeckendorf representation of multiplicative inverses modulo a Fibonacci number

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