scholarly journals On Fibonacci Numbers of Order r Which Are Expressible as Sum of Consecutive Factorial Numbers

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 962
Author(s):  
Eva Trojovská  ◽  
Pavel Trojovský
Keyword(s):  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Initial Values

Let (tn(r))n≥0 be the sequence of the generalized Fibonacci number of order r, which is defined by the recurrence tn(r)=tn−1(r)+⋯+tn−r(r) for n≥r, with initial values t0(r)=0 and ti(r)=1, for all 1≤i≤r. In 2002, Grossman and Luca searched for terms of the sequence (tn(2))n, which are expressible as a sum of factorials. In this paper, we continue this program by proving that, for any ℓ≥1, there exists an effectively computable constant C=C(ℓ)>0 (only depending on ℓ), such that, if (m,n,r) is a solution of tm(r)=n!+(n+1)!+⋯+(n+ℓ)!, with r even, then max{m,n,r}<C. As an application, we solve the previous equation for all 1≤ℓ≤5.

On a system of three difference equations of higher order solved in terms of Lucas and Fibonacci numbers

2020 ◽  
Vol 70 (3) ◽  
pp. 641-656
Author(s):  
Amira Khelifa ◽  
Yacine Halim ◽  
Abderrahmane Bouchair ◽  
Massaoud Berkal
Keyword(s):  
General Solution ◽  
Closed Form ◽  
Difference Equations ◽  
Fibonacci Numbers ◽  
Higher Order ◽  
Real Numbers ◽  
Lucas Numbers ◽  
Rational Difference Equations ◽  
Initial Values ◽  
Fibonacci And Lucas Numbers

AbstractIn this paper we give some theoretical explanations related to the representation for the general solution of the system of the higher-order rational difference equations$$\begin{array}{} \displaystyle x_{n+1} = \dfrac{1+2y_{n-k}}{3+y_{n-k}},\qquad y_{n+1} = \dfrac{1+2z_{n-k}}{3+z_{n-k}},\qquad z_{n+1} = \dfrac{1+2x_{n-k}}{3+x_{n-k}}, \end{array}$$where n, k∈ ℕ0, the initial values x−k, x−k+1, …, x0, y−k, y−k+1, …, y0, z−k, z−k+1, …, z1 and z0 are arbitrary real numbers do not equal −3. This system can be solved in a closed-form and we will see that the solutions are expressed using the famous Fibonacci and Lucas numbers.

scholarly journals On the Sum of Powers of Two k-Fibonacci Numbers which Belongs to the Sequence of k-Lucas Numbers

2016 ◽  
Vol 67 (1) ◽  
pp. 41-46
Author(s):  
Pavel Trojovský
Keyword(s):  
Recurrence Relation ◽  
Initial Conditions ◽  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Lucas Numbers ◽  
Initial Values ◽  
Lucas Sequence ◽  
Positive Integers

Abstract Let k ≥ 1 and denote (Fk,n)n≥0, the k-Fibonacci sequence whose terms satisfy the recurrence relation Fk,n = kFk,n−1 +Fk,n−2, with initial conditions Fk,0 = 0 and Fk,1 = 1. In the same way, the k-Lucas sequence (Lk,n)n≥0 is defined by satisfying the same recurrence relation with initial values Lk,0 = 2 and Lk,1 = k. These sequences were introduced by Falcon and Plaza, who showed many of their properties, too. In particular, they proved that Fk,n+1 + Fk,n−1 = Lk,n, for all k ≥ 1 and n ≥ 0. In this paper, we shall prove that if k ≥ 1 and $F_{k,n + 1}^s + F_{k,n - 1}^s \in \left( {L_{k,m} } \right)_{m \ge 1} $ for infinitely many positive integers n, then s =1.

scholarly journals Some Diophantine Problems Related to k-Fibonacci Numbers

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1047
Author(s):  
Pavel Trojovský ◽  
Štěpán Hubálovský
Keyword(s):  
Recurrence Relation ◽  
Diophantine Equations ◽  
Initial Conditions ◽  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Recursive Relation ◽  
Initial Values ◽  
Lucas Sequence ◽  
Diophantine Problems ◽  
Positive Integers

Let k ≥ 1 be an integer and denote ( F k , n ) n as the k-Fibonacci sequence whose terms satisfy the recurrence relation F k , n = k F k , n − 1 + F k , n − 2 , with initial conditions F k , 0 = 0 and F k , 1 = 1 . In the same way, the k-Lucas sequence ( L k , n ) n is defined by satisfying the same recursive relation with initial values L k , 0 = 2 and L k , 1 = k . The sequences ( F k , n ) n ≥ 0 and ( L k , n ) n ≥ 0 were introduced by Falcon and Plaza, who derived many of their properties. In particular, they proved that F k , n 2 + F k , n + 1 2 = F k , 2 n + 1 and F k , n + 1 2 − F k , n − 1 2 = k F k , 2 n , for all k ≥ 1 and n ≥ 0 . In this paper, we shall prove that if k > 1 and F k , n s + F k , n + 1 s ∈ ( F k , m ) m ≥ 1 for infinitely many positive integers n, then s = 2 . Similarly, that if F k , n + 1 s − F k , n − 1 s ∈ ( k F k , m ) m ≥ 1 holds for infinitely many positive integers n, then s = 1 or s = 2 . This generalizes a Marques and Togbé result related to the case k = 1 . Furthermore, we shall solve the Diophantine equations F k , n = L k , m , F k , n = F n , k and L k , n = L n , k .

scholarly journals Fibonacci Numbers with a Prescribed Block of Digits

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 639 ◽  
Author(s):  
Pavel Trojovský
Keyword(s):  
Lower Bounds ◽  
Diophantine Approximation ◽  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Algebraic Numbers ◽  
Decimal Expansion ◽  
Linear Forms

In this paper, we prove that F 22 = 17711 is the largest Fibonacci number whose decimal expansion is of the form a b … b c … c . The proof uses lower bounds for linear forms in three logarithms of algebraic numbers and some tools from Diophantine approximation.

On the $x$-coordinates of Pell equations which are Fibonacci numbers

2018 ◽  
Vol 122 (1) ◽  
pp. 18 ◽  
Author(s):  
Florian Luca ◽  
Alain Togbé
Keyword(s):  
Positive Integer ◽  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Pell Equation ◽  
Pell Equations

For an integer $d>2$ which is not a square, we show that there is at most one value of the positive integer $x$ participating in the Pell equation $x^2-dy^2=\pm 1$ which is a Fibonacci number.

scholarly journals A curious property of series involving terms of generalized sequences

2000 ◽  
Vol 23 (1) ◽  
pp. 55-63
Author(s):  
Odoardo Brugia ◽  
Piero Filipponi
Keyword(s):  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Lucas Numbers ◽  
Generalized Fibonacci Numbers ◽  
Generalized Lucas Numbers ◽  
Curious Property

Here we are concerned with series involving generalized Fibonacci numbersUn  (p,q)and generalized Lucas numbersVn  (p,q). The aim of this paper is to find triples(p,q,r)for which the seriesUn  (p,q)/rnandVn  (p,q)/rn(forrrunning from 0 to infinity) are unconcerned at the introduction of the factorn. The results established in this paper generalize the known fact that the seriesFn/2n(Fnthenth Fibonacci number) and the seriesnFn/2ngive the same result, namely−2/5.

scholarly journals THE QUOTIENT SET OF -GENERALISED FIBONACCI NUMBERS IS DENSE IN

2017 ◽  
Vol 96 (1) ◽  
pp. 24-29 ◽  
Author(s):  
CARLO SANNA
Keyword(s):  
Number Theory ◽  
Prime Number ◽  
Algebraic Number ◽  
Fibonacci Numbers ◽  
Prime Numbers ◽  
Algebraic Number Theory ◽  
Initial Values ◽  
Quotient Set ◽  
Successive Term

The quotient set of $A\subseteq \mathbb{N}$ is defined as $R(A):=\{a/b:a,b\in A,b\neq 0\}$. Using algebraic number theory in $\mathbb{Q}(\sqrt{5})$, Garcia and Luca [‘Quotients of Fibonacci numbers’, Amer. Math. Monthly, to appear] proved that the quotient set of Fibonacci numbers is dense in the $p$-adic numbers $\mathbb{Q}_{p}$ for all prime numbers $p$. For any integer $k\geq 2$, let $(F_{n}^{(k)})_{n\geq -(k-2)}$ be the sequence of $k$-generalised Fibonacci numbers, defined by the initial values $0,0,\ldots ,0,1$ ($k$ terms) and such that each successive term is the sum of the $k$ preceding terms. We use $p$-adic analysis to generalise the result of Garcia and Luca, by proving that the quotient set of $k$-generalised Fibonacci numbers is dense in $\mathbb{Q}_{p}$ for any integer $k\geq 2$ and any prime number $p$.

scholarly journals Novel Fibonacci and non-Fibonacci structure in the sunflower: results of a citizen science experiment

2016 ◽  
Vol 3 (5) ◽  
pp. 160091 ◽  
Author(s):  
Jonathan Swinton ◽  
Erinma Ochu ◽  
Keyword(s):  
Helianthus Annuus ◽  
Citizen Science ◽  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Reliable Data ◽  
Science Study ◽  
The Third ◽  
Data Subset ◽  
First Time

This citizen science study evaluates the occurrence of Fibonacci structure in the spirals of sunflower ( Helianthus annuus ) seedheads. This phenomenon has competing biomathematical explanations, and our core premise is that observation of both Fibonacci and non-Fibonacci structure is informative for challenging such models. We collected data on 657 sunflowers. In our most reliable data subset, we evaluated 768 clockwise or anticlockwise parastichy numbers of which 565 were Fibonacci numbers, and a further 67 had Fibonacci structure of a predefined type. We also found more complex Fibonacci structures not previously reported in sunflowers. This is the third, and largest, study in the literature, although the first with explicit and independently checkable inclusion and analysis criteria and fully accessible data. This study systematically reports for the first time, to the best of our knowledge, seedheads without Fibonacci structure. Some of these are approximately Fibonacci, and we found in particular that parastichy numbers equal to one less than a Fibonacci number were present significantly more often than those one more than a Fibonacci number. An unexpected further result of this study was the existence of quasi-regular heads, in which no parastichy number could be definitively assigned.

scholarly journals A Note on Closed-Form Representation of Fibonacci Numbers Using Fibonacci Trees

2014 ◽  
Vol 2014 ◽  
pp. 1-3 ◽  
Author(s):  
Indhumathi Raman
Keyword(s):  
Closed Form ◽  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Form Representation

We give a new representation of the Fibonacci numbers. This is achieved using Fibonacci trees. With the help of this representation, the nth Fibonacci number can be calculated without having any knowledge about the previous Fibonacci numbers.

Fibonacci numbers: fun and fundamentals for the slow learner

1970 ◽  
Vol 17 (3) ◽  
pp. 204-208
Author(s):  
Sonja Loftus
Keyword(s):  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Slow Learner ◽  
Slow Learners

The problem of making mathematics more a live and more interesting for the slow learner is a challenge for teachers. Too often the slow learner is asked to repeat and repeat the same material year after year. If we as teachers can find material to stimulate students, it can be fun for us too. This paper describes my experiences with teaching slow learners Fibonacci number a topic new to me.

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