A Quadratic Diophantine Equation Involving Generalized Fibonacci Numbers

Chaves, Ana Paula; Trojovský, Pavel

doi:10.3390/math8061010

Cited by 4 publications

(2 citation statements)

References 18 publications

(22 reference statements)

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“…A recent study by Blankers et al considers the dynamics of Julia sets over hyperbolic numbers [17]. Chaves and Trojovský studied quadratic Diophantine equations using generalized Fibonacci numbers [18]. In their work, they use the fact that this can be thought of in terms of consecutive values of an orbit in a quadratic dynamics related to the Mandelbrot set.…”

Section: Introductionmentioning

confidence: 99%

Visualization of Mandelbrot and Julia Sets of Möbius Transformations

Mork

Ulness

2021

Fractal Fract

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This work reports on a study of the Mandelbrot set and Julia set for a generalization of the well-explored function η(z)=z2+λ. The generalization consists of composing with a fixed Möbius transformation at each iteration step. In particular, affine and inverse Möbius transformations are explored. This work offers a new way of visualizing the Mandelbrot and filled-in Julia sets. An interesting and unexpected appearance of hyperbolic triangles occurs in the structure of the Mandelbrot sets for the case of inverse Möbius transforms. Several lemmas and theorems associated with these types of fractal sets are presented.

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Section: Introductionmentioning

confidence: 99%

Visualization of Mandelbrot and Julia Sets of Möbius Transformations

Mork

Ulness

2021

Fractal Fract

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“…Let k be a fixed positive integer greater than or equal to two. Dafnis, Philippou, and Livieris [10] generalized (1) to the Fibonacci and Lucas numbers of order k (for the definitions of the Fibonacci and Lucas numbers of order k, we refer to [3,11], respectively; see, also, [7,12,13]), deriving the following identity by means of color tiling.…”

Section: Introductionmentioning

confidence: 99%

An Alternating Sum of Fibonacci and Lucas Numbers of Order k

2020

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During the last decade, many researchers have focused on proving identities that reveal the relation between Fibonacci and Lucas numbers. Very recently, one of these identities has been generalized to the case of Fibonacci and Lucas numbers of order k. In the present work, we state and prove a new identity regarding an alternating sum of Fibonacci and Lucas numbers of order k. Our result generalizes recent works in this direction.

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On the Characteristic Polynomial of the Generalized k-Distance Tribonacci Sequences

Trojovský¹

2020

Mathematics

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In 2008, I. Włoch introduced a new generalization of Pell numbers. She used special initial conditions so that this sequence describes the total number of special families of subsets of the set of n integers. In this paper, we prove some results about the roots of the characteristic polynomial of this sequence, but we will consider general initial conditions. Since there are currently several types of generalizations of the Pell sequence, it is very difficult for anyone to realize what type of sequence an author really means. Thus, we will call this sequence the generalized k-distance Tribonacci sequence (Tn(k))n≥0.

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A Quadratic Diophantine Equation Involving Generalized Fibonacci Numbers

Cited by 4 publications

References 18 publications

Visualization of Mandelbrot and Julia Sets of Möbius Transformations

Visualization of Mandelbrot and Julia Sets of Möbius Transformations

An Alternating Sum of Fibonacci and Lucas Numbers of Order k

On the Characteristic Polynomial of the Generalized k-Distance Tribonacci Sequences

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