Generalized Fibonacci Sequences

2000 ◽  
Vol 93 (7) ◽  
pp. 604-606
Author(s):  
Sean Bradley
Keyword(s):  
Fibonacci Sequence ◽  
Infinite Family ◽  
Vast Amount ◽  
Generalized Fibonacci Sequences ◽  
Fibonacci Sequences

Everyone loves the Fibonacci sequence. It is easy to describe, yet it gives rise to a vast amount of substantial mathematics. Physical applications and connections with various branches of mathematics abound. What could be better, unless someone told us that the Fibonacci sequence is but one member of an infinite family of sequences that we could be discussing? The generalization that follows has great potential for student and teacher exploration, as well as discovery, wonder, and amusement.

scholarly journals On Terms of Generalized Fibonacci Sequences which are Powers of their Indexes

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 700 ◽  
Author(s):  
Pavel Trojovský
Keyword(s):  
Diophantine Equation ◽  
Fibonacci Sequence ◽  
Generalized Fibonacci Sequence ◽  
Generalized Fibonacci Sequences ◽  
Fibonacci Sequences

The k-generalized Fibonacci sequence ( F n ( k ) ) n (sometimes also called k-bonacci or k-step Fibonacci sequence), with k ≥ 2 , is defined by the values 0 , 0 , … , 0 , 1 of starting k its terms and such way that each term afterwards is the sum of the k preceding terms. This paper is devoted to the proof of the fact that the Diophantine equation F m ( k ) = m t , with t > 1 and m > k + 1 , has only solutions F 12 ( 2 ) = 12 2 and F 9 ( 3 ) = 9 2 .

Generalization of the Fibonacci Sequence to n Dimensions

1966 ◽  
Vol 18 ◽  
pp. 332-349 ◽  
Author(s):  
George N. Raney
Keyword(s):  
Free Semigroup ◽  
Jacobi Polynomials ◽  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Natural Generalization ◽  
Unimodular Group ◽  
Natural Extensions ◽  
Generalized Fibonacci Sequences ◽  
Fibonacci Sequences

We introduce certain n X n matrices with integral elements that constitute a free semigroup with identity and generate the n-dimensional unimodular group. In terms of these matrices we define a certain sequence of n-dimensional vectors, which we show is the natural generalization to n dimensions of the Fibonacci sequence. Connections between the generalized Fibonacci sequences and certain Jacobi polynomials are found. The various basic identities concerning the Fibonacci numbers are shown to have natural extensions to n dimensions, and in some cases the proofs are rendered quite brief by the use of known theorems on matrices.

MULTIFRACTALITY OF GENERALIZED FIBONACCI PROFILES

Fractals ◽  
1993 ◽  
Vol 01 (03) ◽  
pp. 694-701 ◽  
Author(s):  
A.D. FREITAS ◽  
S. COUTINHO
Keyword(s):  
Ising Model ◽  
Initial Conditions ◽  
Fibonacci Sequence ◽  
Real Space ◽  
Local Magnetization ◽  
Renormalization Group Approach ◽  
Generalized Fibonacci Sequences ◽  
Fibonacci Sequences ◽  
First Time ◽  
Infinite Set

The multifractal properties of a class of one parameter generalized Fibonacci sequences are studied. This class of recursion relations, which is defined by an infinite set of sequences similar to the original Fibonacci’s one, appears for the first time in the study of the Ising model by the real space Migdal-Kadanoff renormalization group approach. The whole set of numbers generated by these equations, when properly arranged over the interval [0,1], gives the exact local magnetization profile of the Ising model on generalized hierarchical lattices at the critical temperature. This profile has a multifractal structure showing that an infinite set of exponents is required to describe how its singularities are distributed. The F(α)-function for the measure defined by the normalized profile is numerically obtained and analyzed. Each value of α characterizes the set of numbers generated by one sequence with arbitrary initial conditions. The exact lower (αmin) and upper (αmax) bounds of the spectra are analytically calculated. For a particular value of the parameter, the original Fibonacci sequence appears, generating a set of numbers diverging with the αmin exponent.

scholarly journals On Some Properties of the Hofstadter–Mertens Function

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Pavel Trojovský
Keyword(s):  
Fibonacci Sequence ◽  
Prime Numbers ◽  
Möbius Function ◽  
Chaotic Sequences ◽  
Mobius Function ◽  
Recursive Sequences ◽  
Fibonacci Sequences

Many mathematicians have been interested in the study of recursive sequences. Among them, a class of “chaotic” sequences are named “meta-Fibonacci sequences.” The main example of meta-Fibonacci sequence was introduced by Hofstadter, and it is called the Q-sequence. Recently, Alkan–Fox–Aybar and the author studied the pattern induced by the connection between the Q-sequence and other known sequences. Here, we continue this program by studying a “Mertens’ version” of the Hofstadter sequence, defined (for x>0) by x↦∑n≤xμnQn, where µ(n) is the Möbius function. In particular, as we shall see, this function encodes many interesting properties which relate prime numbers to “meta-sequences”.

scholarly journals Diophantine Triples and k-Generalized Fibonacci Sequences

2016 ◽  
Vol 41 (3) ◽  
pp. 1449-1465 ◽  
Author(s):  
Clemens Fuchs ◽  
Christoph Hutle ◽  
Florian Luca ◽  
László Szalay
Keyword(s):  
Diophantine Triples ◽  
Generalized Fibonacci Sequences ◽  
Fibonacci Sequences
Sign in / Sign up

Export Citation Format

Share Document