On the distance between generalized Fibonacci numbers

Jhon J. Bravo; Carlos A. Gómez; Florian Luca

Displaying similar documents to “On the distance between generalized Fibonacci numbers”

On the intersection of two distinct $k$ -generalized Fibonacci sequences

Diego Marques (2012)

Mathematica Bohemica

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Let $k \geq 2$ and define $F^{(k)} : = {(F_{n}^{(k)})}_{n \geq 0}$ , the $k$ -generalized Fibonacci sequence whose terms satisfy the recurrence relation $F_{n}^{(k)} = F_{n - 1}^{(k)} + F_{n - 2}^{(k)} + \dots + F_{n - k}^{(k)}$ , with initial conditions $0, 0, \dots, 0, 1$ ( $k$ terms) and such that the first nonzero term is $F_{1}^{(k)} = 1$ . The sequences $F : = F^{(2)}$ and $T : = F^{(3)}$ are the known Fibonacci and Tribonacci sequences, respectively. In 2005, Noe and Post made a conjecture related to the possible solutions of the Diophantine equation $F_{n}^{(k)} = F_{m}^{(ℓ)}$ . In this note, we use transcendental tools to provide a general method for finding the intersections $F^{(k)} \cap F^{(m)}$ which gives...

On the golden number and Fibonacci type sequences

Eugeniusz Barcz (2019)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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The paper presents, among others, the golden number $ϕ$ as the limit of the quotient of neighboring terms of the Fibonacci and Fibonacci type sequence by means of a fixed point of a mapping of a certain interval with the help of Edelstein’s theorem. To demonstrate the equality , where $f_{n}$ is $n$ -th Fibonacci number also the formula from Corollary has been applied. It was obtained using some relationships between Fibonacci and Lucas numbers, which were previously justified.

On square classes in generalized Fibonacci sequences

Zafer Şiar, Refik Keskin (2016)

Acta Arithmetica

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Let P and Q be nonzero integers. The generalized Fibonacci and Lucas sequences are defined respectively as follows: U₀ = 0, U₁ = 1, V₀ = 2, V₁ = P and $U_{n + 1} = P U ₙ + Q U_{n - 1}$ , $V_{n + 1} = P V ₙ + Q V_{n - 1}$ for n ≥ 1. In this paper, when w ∈ 1,2,3,6, for all odd relatively prime values of P and Q such that P ≥ 1 and P² + 4Q > 0, we determine all n and m satisfying the equation Uₙ = wUₘx². In particular, when k|P and k > 1, we solve the equations Uₙ = kx² and Uₙ = 2kx². As a result, we determine all n such that Uₙ = 6x². ...

Some interpretations of the $(k, p)$ -Fibonacci numbers

Natalia Paja, Iwona Włoch (2021)

Commentationes Mathematicae Universitatis Carolinae

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In this paper we consider two parameters generalization of the Fibonacci numbers and Pell numbers, named as the $(k, p)$ -Fibonacci numbers. We give some new interpretations of these numbers. Moreover using these interpretations we prove some identities for the $(k, p)$ -Fibonacci numbers.

An exponential Diophantine equation related to the sum of powers of two consecutive k-generalized Fibonacci numbers

Carlos Alexis Gómez Ruiz, Florian Luca (2014)

Colloquium Mathematicae

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A generalization of the well-known Fibonacci sequence ${F ₙ}_{n \geq 0}$ given by F₀ = 0, F₁ = 1 and $F_{n + 2} = F_{n + 1} + F ₙ$ for all n ≥ 0 is the k-generalized Fibonacci sequence ${F ₙ^{(k)}}_{n \geq - (k - 2)}$ whose first k terms are 0,..., 0, 1 and each term afterwards is the sum of the preceding k terms. For the Fibonacci sequence the formula $F ₙ ² + F ²_{n + 1} ² = F_{2 n + 1}$ holds for all n ≥ 0. In this paper, we show that there is no integer x ≥ 2 such that the sum of the xth powers of two consecutive k-generalized Fibonacci numbers is again a k-generalized Fibonacci number. This...

On the spacing between terms of generalized Fibonacci sequences

Diego Marques (2014)

Colloquium Mathematicae

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For k ≥ 2, the k-generalized Fibonacci sequence $(F ₙ^{(k)}) ₙ$ is defined to have the initial k terms 0,0,...,0,1 and be such that each term afterwards is the sum of the k preceding terms. We will prove that the number of solutions of the Diophantine equation $F ₘ^{(k)} - F ₙ^{(ℓ)} = c > 0$ (under some weak assumptions) is bounded by an effectively computable constant depending only on c.

Fermat $k$ -Fibonacci and $k$ -Lucas numbers

Jhon J. Bravo, Jose L. Herrera (2020)

Mathematica Bohemica

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Using the lower bound of linear forms in logarithms of Matveev and the theory of continued fractions by means of a variation of a result of Dujella and Pethő, we find all $k$ -Fibonacci and $k$ -Lucas numbers which are Fermat numbers. Some more general results are given.

On a generalization of the Pell sequence

Jhon J. Bravo, Jose L. Herrera, Florian Luca (2021)

Mathematica Bohemica

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The Pell sequence ${(P_{n})}_{n = 0}^{\infty}$ is the second order linear recurrence defined by $P_{n} = 2 P_{n - 1} + P_{n - 2}$ with initial conditions $P_{0} = 0$ and $P_{1} = 1$ . In this paper, we investigate a generalization of the Pell sequence called the $k$ -generalized Pell sequence which is generated by a recurrence relation of a higher order. We present recurrence relations, the generalized Binet formula and different arithmetic properties for the above family of sequences. Some interesting identities involving the Fibonacci and generalized Pell numbers...

Some identities involving differences of products of generalized Fibonacci numbers

Curtis Cooper (2015)

Colloquium Mathematicae

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Melham discovered the Fibonacci identity $F_{n + 1} F_{n + 2} F_{n + 6} - F ³_{n + 3} = (- 1) ⁿ F ₙ$ . He then considered the generalized sequence Wₙ where W₀ = a, W₁ = b, and $W ₙ = p W_{n - 1} + q W_{n - 2}$ and a, b, p and q are integers and q ≠ 0. Letting e = pab - qa² - b², he proved the following identity: $W_{n + 1} W_{n + 2} W_{n + 6} - W ³_{n + 3} = e q^{n + 1} (p ³ W_{n + 2} - q ² W_{n + 1})$ . There are similar differences of products of Fibonacci numbers, like this one discovered by Fairgrieve and Gould: $F ₙ F_{n + 4} F_{n + 5} - F ³_{n + 3} = {(- 1)}^{n + 1} F_{n + 6}$ . We prove similar identities. For example, a generalization of Fairgrieve and Gould’s identity is $W ₙ W_{n + 4} W_{n + 5} - W ³_{n + 3} = e q ⁿ (p ³ W_{n + 4} - q W_{n + 5})$ .

Piatetski-Shapiro sequences via Beatty sequences

Lukas Spiegelhofer (2014)

Acta Arithmetica

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Integer sequences of the form $⌊ n^{c} ⌋$ , where 1 < c < 2, can be locally approximated by sequences of the form ⌊nα+β⌋ in a very good way. Following this approach, we are led to an estimate of the difference $\sum_{n \leq x} φ (⌊ n^{c} ⌋) - 1 / c \sum_{n \leq x^{c}} φ (n) n^{1 / c - 1}$ , which measures the deviation of the mean value of φ on the subsequence $⌊ n^{c} ⌋$ from the expected value, by an expression involving exponential sums. As an application we prove that for 1 < c ≤ 1.42 the subsequence of the Thue-Morse sequence indexed by $⌊ n^{c} ⌋$ attains both of its values with...

Primefree shifted Lucas sequences

Lenny Jones (2015)

Acta Arithmetica

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We say a sequence $= {(s ₙ)}_{n \geq 0}$ is primefree if |sₙ| is not prime for all n ≥ 0, and to rule out trivial situations, we require that no single prime divides all terms of . In this article, we focus on the particular Lucas sequences of the first kind, $_{a} = {(u ₙ)}_{n \geq 0}$ , defined by u₀ = 0, u₁ = 1, and uₙ = aun-1 + un-2 for n≥2, where a is a fixed integer. More precisely, we show that for any integer a, there exist infinitely many integers k such that both of the shifted sequences $_{a} \pm k$ are simultaneously primefree. This...

Explicit algebraic dependence formulae for infinite products related with Fibonacci and Lucas numbers

Hajime Kaneko, Takeshi Kurosawa, Yohei Tachiya, Taka-aki Tanaka (2015)

Acta Arithmetica

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Let d ≥ 2 be an integer. In 2010, the second, third, and fourth authors gave necessary and sufficient conditions for the infinite products $\prod_{\binom{k = 1}{U_{d^{k}} \neq - a_{i}}}^{\infty} (1 + (a_{i}) / (U_{d^{k}}))$ (i=1,...,m) or $\prod_{\binom{k = 1}{V_{d^{k}} \neq - a_{i}}}^{\infty} (1 + (a_{i}) (V_{d^{k}})$ (i=1,...,m) to be algebraically dependent, where $a_{i}$ are non-zero integers and $U_{n}$ and $V_{n}$ are generalized Fibonacci numbers and Lucas numbers, respectively. The purpose of this paper is to relax the condition on the non-zero integers $a_{1}, . . ., a_{m}$ to non-zero real algebraic numbers, which gives new cases where the infinite products above are algebraically...

Lucas factoriangular numbers

Bir Kafle, Florian Luca, Alain Togbé (2020)

Mathematica Bohemica

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We show that the only Lucas numbers which are factoriangular are $1$ and $2$ .

A new proof of the $q$ -Dixon identity

Victor J. W. Guo (2018)

Czechoslovak Mathematical Journal

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We give a new and elementary proof of Jackson’s terminating $q$ -analogue of Dixon’s identity by using recurrences and induction.

Root location for the characteristic polynomial of a Fibonacci type sequence

Zhibin Du, Carlos Martins da Fonseca (2023)

Czechoslovak Mathematical Journal

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We analyse the roots of the polynomial $x^{n} - p x^{n - 1} - q x - 1$ for $p ⩾ q ⩾ 1$ . This is the characteristic polynomial of the recurrence relation $F_{k, p, q} (n) = p F_{k, p, q} (n - 1) + q F_{k, p, q} (n - k + 1) + F_{k, p, q} (n - k)$ for $n ⩾ k$ , which includes the relations of several particular sequences recently defined. In the end, a matricial representation for such a recurrence relation is provided.

Interpolating sequences in the unit ball of $C^{n}$

N. Pandeski (1985)

Matematički Vesnik

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