Displaying similar documents to “A generalization of the Cassini formula”

An inequality for Fibonacci numbers

Horst Alzer, Florian Luca (2022)

Mathematica Bohemica

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We extend an inequality for Fibonacci numbers published by P. G. Popescu and J. L. Díaz-Barrero in 2006.

Gelin-Cesáro identities for Fibonacci and Lucas quaternions

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.

Binomials transformation formulae for scaled Fibonacci numbers

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.

On terms of linear recurrence sequences with only one distinct block of digits

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.

Leonardo Fibonacci and Abbaco Culture. A Proposal to Invert the Roles

Jens Høyrup (2005)

Revue d'histoire des mathématiques

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Since long it has been regarded as an obvious fact in need of no argument that the mathematics of the Italian abbacus school was taken over from Leonardo Fibonacci’s . What does look like an argument is that an abbacus book from the outgoing 13th century (apparently the earliest extant specimen) claims to be made “according to the opinion” of Fibonacci. Close analysis of the text reveals, however, that everything basic is independent of Fibonacci, while the indubitable borrowings from...

Preface

J. Berstel, T. Harju, J. Karhumäki (2008)

RAIRO - Theoretical Informatics and Applications

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Determinants and inverses of circulant matrices with complex Fibonacci numbers

Ercan Altınışık, N. Feyza Yalçın, Şerife Büyükköse (2015)

Special Matrices

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Let ℱn = circ (︀F*1 , F*2, . . . , F*n︀ be the n×n circulant matrix associated with complex Fibonacci numbers F*1, F*2, . . . , F*n. In the present paper we calculate the determinant of ℱn in terms of complex Fibonacci numbers. Furthermore, we show that ℱn is invertible and obtain the entries of the inverse of ℱn in terms of complex Fibonacci numbers.

The positivity problem for fourth order linear recurrence sequences is decidable

Pinthira Tangsupphathawat, Narong Punnim, Vichian Laohakosol (2012)

Colloquium Mathematicae

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The problem whether each element of a sequence satisfying a fourth order linear recurrence with integer coefficients is nonnegative, referred to as the Positivity Problem for fourth order linear recurrence sequence, is shown to be decidable.