Generalization of an identity involving the generalized Fibonacci numbers and its applications.
Mohammad Farrokhi, D.G. (2009)
Integers
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Mohammad Farrokhi, D.G. (2009)
Integers
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Ji-Cai Liu (2017)
Czechoslovak Mathematical Journal
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Euler's pentagonal number theorem was a spectacular achievement at the time of its discovery, and is still considered to be a beautiful result in number theory and combinatorics. In this paper, we obtain three new finite generalizations of Euler's pentagonal number theorem.
Ahmet Daşdemir (2019)
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
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To date, many identities of different quaternions, including the Fibonacci and Lucas quaternions, have been investigated. In this study, we present Gelin-Cesáro identities for Fibonacci and Lucas quaternions. The identities are a worthy addition to the literature. Moreover, we give Catalan's identity for the Lucas quaternions.
István Mező, Ayhan Dil (2009)
Open Mathematics
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In this paper we use the Euler-Seidel method for deriving new identities for hyperharmonic and r-Stirling numbers. The exponential generating function is determined for hyperharmonic numbers, which result is a generalization of Gosper’s identity. A classification of second order recurrence sequences is also given with the help of this method.
Alexey Stakhov (2012)
Visual Mathematics
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William D. Banks, Florian Luca (2005)
Acta Arithmetica
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Kiliç, Emrah, Tan, Elif (2010)
Integers
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William D. Banks, John B. Friedlander, Florian Luca, Francesco Pappalardi, Igor E. Shparlinski (2006)
Acta Arithmetica
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Florian Luca, Pantelimon Stănică (2007)
Acta Arithmetica
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Noe, Tony D., Vos Post, Jonathan (2005)
Journal of Integer Sequences [electronic only]
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Tasoev, B.G. (1999)
Vladikavkazskiĭ Matematicheskiĭ Zhurnal
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Edyta Hetmaniok, Bożena Piątek, Roman Wituła (2017)
Open Mathematics
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The aim of the paper is to present the binomial transformation formulae of Fibonacci numbers scaled by complex multipliers. Many of these new and nontrivial relations follow from the fundamental properties of the so-called delta-Fibonacci numbers defined by Wituła and Słota. The paper contains some original relations connecting the values of delta-Fibonacci numbers with the respective values of Chebyshev polynomials of the first and second kind.
Hao Pan, Zhi-Wei Sun (2006)
Acta Arithmetica
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Florian Luca (2011)
Bulletin of the Polish Academy of Sciences. Mathematics
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Zhang, Tianping, Ma, Yuankui (2005)
Journal of Integer Sequences [electronic only]
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Światosław R. Gal (2001)
Colloquium Mathematicae
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A closed form formula (generating function) for the Euler characteristic of the configuration space of n particles in a simplicial complex is given.
Vera W. de Spinadel (1999)
Visual Mathematics
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Diego Marques, Alain Togbé (2011)
Colloquium Mathematicae
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In 2000, Florian Luca proved that F₁₀ = 55 and L₅ = 11 are the largest numbers with only one distinct digit in the Fibonacci and Lucas sequences, respectively. In this paper, we find terms of a linear recurrence sequence with only one block of digits in its expansion in base g ≥ 2. As an application, we generalize Luca's result by finding the Fibonacci and Lucas numbers with only one distinct block of digits of length up to 10 in its decimal expansion.