Generalization of an Identity Involving the Generalized Fibonacci Numbers and Its Applications

Integers ◽  
2009 ◽  
Vol 9 (5) ◽  
Author(s):  
D. G. Mohammad Farrokhi
Keyword(s):  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Sufficient Condition ◽  
Generalized Fibonacci Numbers ◽  
Generalized Fibonacci Sequence

AbstractWe first generalize an identity involving the generalized Fibonacci numbers and then apply it to establish some general identities concerning special sums. We also give a sufficient condition on a generalized Fibonacci sequence {

scholarly journals Some Properties of Generalized Fibonacci Numbers: Identities, Recurrence Properties and Closed Forms of the Sum Formulas ∑nk=0 xkWmk+j

2021 ◽  
pp. 11-38
Author(s):  
Yüksel Soykan
Keyword(s):  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Lucas Numbers ◽  
Generalized Fibonacci Numbers ◽  
Generalized Fibonacci Sequence ◽  
Summation Formulas ◽  
Special Cases

In this paper, closed forms of the summation formulas ∑nk=0 xkWmk+j for generalized Fibonacci numbers are presented. As special cases, we give summation formulas of Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas numbers. We present the proofs to indicate how these formulas, in general, were discovered. Of course, all the listed formulas may be proved by induction, but that method of proof gives no clue about their discovery. Moreover, we give some identities and recurrence properties of generalized Fibonacci sequence.

scholarly journals On the problem of Pillai with k-generalized Fibonacci numbers and powers of 3

2020 ◽  
Vol 16 (07) ◽  
pp. 1643-1666
Author(s):  
Mahadi Ddamulira ◽  
Florian Luca
Keyword(s):  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Fibonacci Sequence ◽  
Generalized Fibonacci Numbers ◽  
Generalized Fibonacci Sequence ◽  
Tribonacci Numbers

For an integer [Formula: see text], let [Formula: see text] be the [Formula: see text]-generalized Fibonacci sequence which starts with [Formula: see text] (a total of [Formula: see text] terms) and for which each term afterwards is the sum of the [Formula: see text] preceding terms. In this paper, we find all integers [Formula: see text] with at least two representations as a difference between a [Formula: see text]-generalized Fibonacci number and a power of [Formula: see text]. This paper continues the previous work of the first author for the Fibonacci numbers, and for the Tribonacci numbers.

scholarly journals Motzkin’s Maximal Density and Related Chromatic Numbers

2018 ◽  
Vol 13 (1) ◽  
pp. 27-45
Author(s):  
Anshika Srivastava ◽  
Ram Krishna Pandey ◽  
Om Prakash
Keyword(s):  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Chromatic Numbers ◽  
Distance Graphs ◽  
Maximal Density ◽  
Generalized Fibonacci Numbers ◽  
Generalized Fibonacci Sequence ◽  
Odd Order ◽  
Upper Density ◽  
Positive Integers

Abstract This paper concerns the problem of determining or estimating the maximal upper density of the sets of nonnegative integers S whose elements do not differ by an element of a given set M of positive integers. We find some exact values and some bounds for the maximal density when the elements of M are generalized Fibonacci numbers of odd order. The generalized Fibonacci sequence of order r is a generalization of the well known Fibonacci sequence, where instead of starting with two predetermined terms, we start with r predetermined terms and each term afterwards is the sum of r preceding terms. We also derive some new properties of the generalized Fibonacci sequence of order r. Furthermore, we discuss some related coloring parameters of distance graphs generated by the set M.

scholarly journals On Fibonacci Numbers and its Applications

2019 ◽  
Vol 8 (12) ◽  
pp. 453-455
Keyword(s):  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Generalized Fibonacci Sequence

In this article, we explore the representation of the product of k consecutive Fibonacci numbers as the sum of kth power of Fibonacci numbers. We also present a formula for finding the coefficients of the Fibonacci numbers appearing in this representation. Finally, we extend the idea to the case of generalized Fibonacci sequence and also, we produce another formula for finding the coefficients of Fibonacci numbers appearing in the representation of three consecutive Fibonacci numbers as a particular case. Also, we point out some amazing applications of Fibonacci numbers.

scholarly journals ON THE LARGEST PRIME FACTOR OF THE k-FIBONACCI NUMBERS

2013 ◽  
Vol 09 (05) ◽  
pp. 1351-1366 ◽  
Author(s):  
JHON J. BRAVO ◽  
FLORIAN LUCA
Keyword(s):  
Prime Factor ◽  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Generalized Fibonacci Sequence

Let P(m) denote the largest prime factor of an integer m ≥ 2, and put P(0) = P(1) = 1. For an integer k ≥ 2, let [Formula: see text] be the k-generalized Fibonacci sequence which starts with 0, …, 0, 1 (k terms) and each term afterwards is the sum of the k preceding terms. Here, we show that if n ≥ k+2, then [Formula: see text]. Furthermore, we determine all the k-Fibonacci numbers [Formula: see text] whose largest prime factor is less than or equal to 7.

scholarly journals Curious Generalized Fibonacci Numbers

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2588
Author(s):  
Jose L. Herrera ◽  
Jhon J. Bravo ◽  
Carlos A. Gómez
Keyword(s):  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Generalized Fibonacci Numbers

A generalization of the well-known Fibonacci sequence is the k−Fibonacci sequence whose first k terms are 0,…,0,1 and each term afterwards is the sum of the preceding k terms. In this paper, we find all k-Fibonacci numbers that are curious numbers (i.e., numbers whose base ten representation have the form a⋯ab⋯ba⋯a). This work continues and extends the prior result of Trojovský, who found all Fibonacci numbers with a prescribed block of digits, and the result of Alahmadi et al., who searched for k-Fibonacci numbers, which are concatenation of two repdigits.

scholarly journals Generalized Fibonacci Numbers and Music

2018 ◽  
Vol 14 (1) ◽  
pp. 7564-7579
Author(s):  
Anthony G Shannon ◽  
Irina Klamka ◽  
Robert van Gend
Keyword(s):  
Golden Ratio ◽  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Generalized Fibonacci Numbers

Mathematics and music have well documented historical connections. Just as the ordinary Fibonacci numbers have links with the golden ratio, this paper considers generalized Fibonacci numbers developed from generalizations of the golden ratio. It is well known that the Fibonacci sequence of numbers underlie certain musical intervals and compositions but to what extent are these connections accidental or structural, coincidental or natural and do generalized Fibonacci numbers share any of these connections?

scholarly journals Generalized Fibonacci Numbers, Cosmological Analogies, and an Invariant

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 200
Author(s):  
Valerio Faraoni ◽  
Farah Atieh
Keyword(s):  
General Relativity ◽  
Fibonacci Numbers ◽  
Friedmann Equation ◽  
Fibonacci Sequence ◽  
Generalized Fibonacci Numbers ◽  
Isotropic Cosmology ◽  
Spatially Homogeneous

Continuous generalizations of the Fibonacci sequence satisfy ODEs that are formal analogues of the Friedmann equation describing a spatially homogeneous and isotropic cosmology in general relativity. These analogies are presented together with their Lagrangian and Hamiltonian formulations and with an invariant of the Fibonacci sequence.

On the zero-multiplicity of the k-generalized Fibonacci sequence

2020 ◽  
Vol 26 (11-12) ◽  
pp. 1564-1578
Author(s):  
Jonathan García ◽  
Carlos A. Gómez ◽  
Florian Luca
Keyword(s):  
Fibonacci Sequence ◽  
Generalized Fibonacci Sequence
Sign in / Sign up

Export Citation Format

Share Document