Fibonacci Numbers and Continued Fractions

2002 ◽  
pp. 89-123
Author(s):  
Nicolai N. Vorobiew
Keyword(s):  
Continued Fractions ◽  
Fibonacci Numbers

scholarly journals ARBITRARY ORDER RECURSIVE SEQUENCES AND ASSOCIATED CONTINUED FRACTIONS

2017 ◽  
Vol 13 (2) ◽  
pp. 7147-7154
Author(s):  
Anthony G Shannon ◽  
Charles K Cook b ◽  
Rebecca A. Hillman c
Keyword(s):  
Continued Fractions ◽  
Arbitrary Order ◽  
Fibonacci Numbers ◽  
Recursive Sequences ◽  
Essential Idea ◽  
Associated Continued Fractions ◽  
Generalized Continued Fractions

The essential idea in this paper it to generalize and synthesize some of the pioneering ideas of Bernstein, Lucas and Horadam on generalizations of basic and primordial Fibonacci numbers and their arbitrary order generalizations and their relation to generalized continued fractions with matrices as the unifying elements.

scholarly journals Closed Form Continued Fraction Expansions of Special Quadratic Irrationals

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Daniel Fishman ◽  
Steven J. Miller
Keyword(s):  
Positive Integer ◽  
Closed Form ◽  
Recurrence Relation ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Fibonacci Numbers ◽  
Continued Fraction Expansions

We derive closed form expressions for the continued fractions of powers of certain quadratic surds. Specifically, consider the recurrence relation with , , a positive integer, and (note that gives the Fibonacci numbers). Let . We find simple closed form continued fraction expansions for for any integer by exploiting elementary properties of the recurrence relation and continued fractions.

Fibonacci numbers in generalized Pell sequences

2020 ◽  
Vol 70 (5) ◽  
pp. 1057-1068
Author(s):  
Jhon J. Bravo ◽  
Jose L. Herrera
Keyword(s):  
Lower Bounds ◽  
Continued Fractions ◽  
Fibonacci Numbers ◽  
Independent Interest ◽  
Linear Forms In Logarithms ◽  
Algebraic Numbers ◽  
Linear Forms ◽  
Pell Numbers

AbstractIn this paper, by using lower bounds for linear forms in logarithms of algebraic numbers and the theory of continued fractions, we find all Fibonacci numbers that appear in generalized Pell sequences. Some interesting estimations involving generalized Pell numbers, that we believe are of independent interest, are also deduced. This paper continues a previous work that searched for Fibonacci numbers in the Pell sequence.

scholarly journals Repdigits as Product of Terms of k-Bonacci Sequences

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 682
Author(s):  
Petr Coufal ◽  
Pavel Trojovský
Keyword(s):  
Recent Work ◽  
Continued Fractions ◽  
Diophantine Equations ◽  
Reduction Method ◽  
Fibonacci Numbers ◽  
Decimal Expansion ◽  
Generalized Fibonacci Numbers ◽  
Initial Values ◽  
Transcendental Method

For any integer k≥2, the sequence of the k-generalized Fibonacci numbers (or k-bonacci numbers) is defined by the k initial values F−(k−2)(k)=⋯=F0(k)=0 and F1(k)=1 and such that each term afterwards is the sum of the k preceding ones. In this paper, we search for repdigits (i.e., a number whose decimal expansion is of the form aa…a, with a∈[1,9]) in the sequence (Fn(k)Fn(k+m))n, for m∈[1,9]. This result generalizes a recent work of Bednařík and Trojovská (the case in which (k,m)=(2,1)). Our main tools are the transcendental method (for Diophantine equations) together with the theory of continued fractions (reduction method).

Linear recurrences associated to rays in Pascal’s triangle and combinatorial identities

2014 ◽  
Vol 64 (2) ◽  
Author(s):  
Hacène Belbachir ◽  
Takao Komatsu ◽  
László Szalay
Keyword(s):  
Recurrence Relation ◽  
Generating Function ◽  
Finite Differences ◽  
Continued Fractions ◽  
Fibonacci Numbers ◽  
Combinatorial Identities ◽  
Bivariate Polynomials ◽  
Linear Recurrences ◽  
Pascal’S Triangle ◽  
Pascal's Triangle

AbstractOur main purpose is to describe the recurrence relation associated to the sum of diagonal elements laying along a finite ray crossing Pascal’s triangle. We precise the generating function of the sequence of described sums. We also answer a question of Horadam posed in his paper [Chebyshev and Pell connections, Fibonacci Quart. 43 (2005), 108–121]. Further, using Morgan-Voyce sequence, we establish the nice identity $F_{n + 1} - iF_n = i^n \sum\limits_{k = 0}^n {(_{2k}^{n + k} )( - 2 - i)^k } $ of Fibonacci numbers, where i is the imaginary unit. Finally, connections to continued fractions, bivariate polynomials and finite differences are given.

scholarly journals A Lochs-Type Approach via Entropy in Comparing the Efficiency of Different Continued Fraction Algorithms

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 255
Author(s):  
Dan Lascu ◽  
Gabriela Ileana Sebe
Keyword(s):  
Dynamical Systems ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Unit Interval ◽  
Probability Measures ◽  
Absolutely Continuous ◽  
Continued Fraction Expansions ◽  
Absolutely Continuous Invariant

We investigate the efficiency of several types of continued fraction expansions of a number in the unit interval using a generalization of Lochs theorem from 1964. Thus, we aim to compare the efficiency by describing the rate at which the digits of one number-theoretic expansion determine those of another. We study Chan’s continued fractions, θ-expansions, N-continued fractions, and Rényi-type continued fractions. A central role in fulfilling our goal is played by the entropy of the absolutely continuous invariant probability measures of the associated dynamical systems.

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