scholarly journals Some generalized results related to Fibonacci sequence

2021 ◽  
Vol 40 (3) ◽  
pp. 605-617
Author(s):  
Neeraj Kumar Paul ◽  
Helen K. Saikia
Keyword(s):  
Fibonacci Number ◽  
Fibonacci Sequence

Cassini's identity states that for the nth Fibonacci number Fn+1Fn-1-Fn2=(-1)n. We generalize Fibonacci sequence in terms of the number of sequences. Fibonacci sequence is the particular case of generating only one sequence. This generalization is used to generalize Cassini’s identity. Moreover we prove few more results which can be seen as generalized form of the results which hold for Fibonacci sequence.

Spiral growth inNephrolepidina: evidence of “golden selection”

Paleobiology ◽  
10.1666/12057 ◽  
2014 ◽  
Vol 40 (2) ◽  
pp. 151-161 ◽  
Author(s):  
Andrea Benedetti
Keyword(s):  
Fibonacci Number ◽  
Fibonacci Sequence ◽  
Spiral Growth ◽  
Regular Growth ◽  
Mean Values ◽  
Evolutionary Selection ◽  
Golden Angle ◽  
New Type ◽  
New Evidence ◽  
Interpretative Model

Examination of the neanic apparatuses of known populations ofNephrolepidina praemarginata,N. morgani, andN. tournouerireveals that the equatorial chamberlets are arranged in spirals, along the direction of connection of the oblique stolons, giving the optical effect of intersecting curves. InN. praemarginatacommonly 34 left- and right-oriented primary spirals occur from the first annulus to the periphery, 21 secondary spirals from the third to fifth annulus, 13 ternary spirals from the fifth to eighth annulus, following the Fibonacci sequence.The number of the spirals increases in larger specimens and in more embracing morphotypes, and especially in trybliolepidine specimens; the secondary and ternary spirals from the investigatedN. praemarginatatoN. tournoueripopulations tend to start from more distal annuli. An interpretative model of the spiral growth ofNephrolepidinais attempted.The angle formed by the basal annular stolon and distal oblique stolon in equatorial chamberlets ranges from 122° inN. praemarginatato mean values close to the golden angle (137.5°) inN. tournoueri.The increase in the Fibonacci number of spirals during the evolution of the lineage, along with the disposition of the stolons between contiguous equatorial chamberlets, provides new evidence of evolutionary selection for specimens with optimally packed chamberlets.Natural selection favors individuals with the most regular growth, which fills the equatorial space more efficiently, thus allowing these individuals to reach the adult stage faster. We refer to this new type of selection as “golden selection.”

On the local behavior of the order of appearance in the Fibonacci sequence

2014 ◽  
Vol 10 (04) ◽  
pp. 915-933 ◽  
Author(s):  
Florian Luca ◽  
Carl Pomerance
Keyword(s):  
Positive Integer ◽  
Rational Function ◽  
Fibonacci Number ◽  
Fibonacci Sequence ◽  
Asymptotic Density ◽  
Infinitely Many Solutions ◽  
Local Behavior ◽  
Sieve Methods ◽  
Density Zero

Let z(N) be the order of appearance of N in the Fibonacci sequence. This is the smallest positive integer k such that N divides the k th Fibonacci number. We show that each of the six total possible orderings among z(N), z(N + 1), z(N + 2) appears infinitely often. We also show that for each nonzero even integer c and many odd integers c the equation z(N) = z(N + c) has infinitely many solutions N, but the set of solutions has asymptotic density zero. The proofs use a result of Corvaja and Zannier on the height of a rational function at 𝒮-unit points as well as sieve methods.

scholarly journals On Integer Numbers with Locally Smallest Order of Appearance in the Fibonacci Sequence

2011 ◽  
Vol 2011 ◽  
pp. 1-4 ◽  
Author(s):  
Diego Marques
Keyword(s):  
Natural Number ◽  
Fibonacci Number ◽  
Fibonacci Sequence ◽  
Natural Numbers

LetFnbe thenth Fibonacci number. The order of appearancez(n)of a natural numbernis defined as the smallest natural numberksuch thatndividesFk. For instance, for alln=Fm≥5, we havez(n−1)>z(n)<z(n+1). In this paper, we will construct infinitely many natural numbers satisfying the previous inequalities and which do not belong to the Fibonacci sequence.

scholarly journals On Diophantine Equations Related to Order of Appearance in Fibonacci Sequence

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1073 ◽  
Author(s):  
Pavel Trojovský
Keyword(s):  
Natural Number ◽  
Diophantine Equations ◽  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Fibonacci Sequence ◽  
Prime Numbers ◽  
All Solutions ◽  
Adic Valuation

Let F n be the nth Fibonacci number. Order of appearance z ( n ) of a natural number n is defined as smallest natural number k, such that n divides F k . In 1930, Lehmer proved that all solutions of equation z ( n ) = n ± 1 are prime numbers. In this paper, we solve equation z ( n ) = n + ℓ for | ℓ | ∈ { 1 , … , 9 } . Our method is based on the p-adic valuation of Fibonacci numbers.

Representing Numbers Using Fibonacci Variants

2015 ◽  
Author(s):  
Stephen K. Lucas
Keyword(s):  
Natural Number ◽  
Greedy Algorithm ◽  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Fibonacci Sequence ◽  
Fixed Number ◽  
Natural Numbers ◽  
Form Versus ◽  
Entire Number

This chapter introduces the Zeckendorf representation of a Fibonacci sequence, a form of a natural number which can be easily found using a greedy algorithm: given a number, subtract the largest Fibonacci number less than or equal to it, and repeat until the entire number is used up. This chapter first compares the efficiency of representing numbers using Zeckendorf form versus traditional binary with a fixed number of digits and shows when Zeckendorf form is to be preferred. It also shows what happens when variants of Zeckendorf form are used. Not only can natural numbers as be presented sums of Fibonacci numbers, but arithmetic can also be done with them directly in Zeckendorf form. The chapter includes a survey of past approaches to Zeckendorf representation arithmetic, as well as some improvements.

scholarly journals Some properties of the generalized (p,q)- Fibonacci-Like number

2018 ◽  
Vol 189 ◽  
pp. 03028
Author(s):  
Alongkot Suvarnamani
Keyword(s):  
Generating Function ◽  
Real World ◽  
Open Problem ◽  
Basic Concept ◽  
Fibonacci Number ◽  
Fibonacci Sequence ◽  
Lucas Number ◽  
Generalized Fibonacci Sequence ◽  
Lucas Sequence ◽  
Real World Problems

For the real world problems, we use some knowledge for explain or solving them. For example, some mathematicians study the basic concept of the generalized Fibonacci sequence and Lucas sequence which are the (p,q) – Fibonacci sequence and the (p,q) – Lucas sequence. Such as, Falcon and Plaza showed some results of the k-Fibonacci sequence. Then many researchers showed some results of the k-Fibonacci- Like number. Moreover, Suvarnamani and Tatong showed some results of the (p, q) - Fibonacci number. They found some properties of the (p,q) – Fibonacci number and the (p,q) – Lucas number. There are a lot of open problem about them. In this paper, we studied about the generalized (p,q)- Fibonacci-Like sequence. We establish properties like Catalan’s identity, Cassini’s identity, Simpson’s identity, d’Ocagne’s identity and Generating function for the generalized (p,q)-Fibonacci-Like number by using the Binet formulas. However, all results which be showed in this paper, are generalized of the (p,q) – Fibonacci-like number and the (p,q) – Fibonacci number. Corresponding author: [email protected]

scholarly journals The Proof of a Conjecture Related to Divisibility Properties of z(n)

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2638
Author(s):  
Eva Trojovská ◽  
Kandasamy Venkatachalam
Keyword(s):  
Positive Integer ◽  
Fibonacci Number ◽  
Fibonacci Sequence ◽  
Natural Density ◽  
Almost All ◽  
Divisibility Properties ◽  
Positive Integers

The order of appearance of n (in the Fibonacci sequence) z(n) is defined as the smallest positive integer k for which n divides the k—the Fibonacci number Fk. Very recently, Trojovský proved that z(n) is an even number for almost all positive integers n (in the natural density sense). Moreover, he conjectured that the same is valid for the set of integers n ≥ 1 for which the integer 4 divides z(n). In this paper, among other things, we prove that for any k ≥ 1, the number z(n) is divisible by 2k for almost all positive integers n (in particular, we confirm Trojovský’s conjecture).

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.

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

scholarly journals Note on some representations of general solutions to homogeneous linear difference equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Stevo Stević ◽  
Bratislav Iričanin ◽  
Witold Kosmala ◽  
Zdeněk Šmarda
Keyword(s):  
General Solution ◽  
Difference Equation ◽  
Difference Equations ◽  
Fibonacci Sequence ◽  
Linear Difference Equation ◽  
Linear Difference Equations ◽  
Second Order Difference Equation ◽  
Constant Coefficients ◽  
Order Difference Equation ◽  
Homogeneous Linear

Abstract It is known that every solution to the second-order difference equation $x_{n}=x_{n-1}+x_{n-2}=0$ x n = x n − 1 + x n − 2 = 0 , $n\ge 2$ n ≥ 2 , can be written in the following form $x_{n}=x_{0}f_{n-1}+x_{1}f_{n}$ x n = x 0 f n − 1 + x 1 f n , where $f_{n}$ f n is the Fibonacci sequence. Here we find all the homogeneous linear difference equations with constant coefficients of any order whose general solution have a representation of a related form. We also present an interesting elementary procedure for finding a representation of general solution to any homogeneous linear difference equation with constant coefficients in terms of the coefficients of the equation, initial values, and an extension of the Fibonacci sequence. This is done for the case when all the roots of the characteristic polynomial associated with the equation are mutually different, and then it is shown that such obtained representation also holds in other cases. It is also shown that during application of the procedure the extension of the Fibonacci sequence appears naturally.

Sign in / Sign up

Export Citation Format

Share Document