Interpolating sequences in the unit ball of
N. Pandeski (1985)
Matematički Vesnik
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N. Pandeski (1985)
Matematički Vesnik
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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 , 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. ...
Jhon J. Bravo, Carlos A. Gómez, Florian Luca (2015)
Colloquium Mathematicae
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For an integer k ≥ 2, let be the k-Fibonacci sequence which starts with 0,..., 0,1 (k terms) and each term afterwards is the sum of the k preceding terms. This paper completes a previous work of Marques (2014) which investigated the spacing between terms of distinct k-Fibonacci sequences.
Diego Marques (2012)
Mathematica Bohemica
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Let and define , the -generalized Fibonacci sequence whose terms satisfy the recurrence relation , with initial conditions ( terms) and such that the first nonzero term is . The sequences and 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 . In this note, we use transcendental tools to provide a general method for finding the intersections which gives...
Lukas Spiegelhofer (2014)
Acta Arithmetica
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Integer sequences of the form , 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 , which measures the deviation of the mean value of φ on the subsequence 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 attains both of its values with...
Lenny Jones (2015)
Acta Arithmetica
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We say a sequence 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, , 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 are simultaneously primefree. This...
Erzsébet Orosz (2006)
Acta Mathematica Universitatis Ostraviensis
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The purpose of this paper is to prove that the common terms of linear recurrences and have at most common terms if , and have at most three common terms if where and are fixed positive integers and is a prime, such that neither nor is perfect square, further are nonzero integers satisfying the equations and .
Roswitha Hofer, Olivier Ramaré (2016)
Acta Arithmetica
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We consider sequences modulo one that are generated using a generalized polynomial over the real numbers. Such polynomials may also involve the integer part operation [·] additionally to addition and multiplication. A well studied example is the (nα) sequence defined by the monomial αx. Their most basic sister, , is less investigated. So far only the uniform distribution modulo one of these sequences is resolved. Completely new, however, are the discrepancy results proved in this paper....
Eugeniusz Barcz (2019)
Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia
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The paper presents, among others, the golden number 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 is -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.
Zhi-Hong Sun (2022)
Czechoslovak Mathematical Journal
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We systematically investigate the expressions and congruences for both a one-parameter family as well as a two-parameter family of sequences.
Zhibin Du, Carlos Martins da Fonseca (2023)
Czechoslovak Mathematical Journal
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We analyse the roots of the polynomial for . This is the characteristic polynomial of the recurrence relation for , which includes the relations of several particular sequences recently defined. In the end, a matricial representation for such a recurrence relation is provided.
Diego Marques (2014)
Colloquium Mathematicae
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For k ≥ 2, the k-generalized Fibonacci sequence 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 (under some weak assumptions) is bounded by an effectively computable constant depending only on c.