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Additional Fibonacci-Bernoulli relations
Researches in Mathematics, 2022 We continue our study on relationships between Fibonacci (Lucas) numbers and Bernoulli numbers and polynomials. The derivations of our results are based on functional equations for the respective generating functions, which in our case are combinations ...
K. Adegoke, R. Frontczak, T.P. Goy
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Repdigits as difference of two Fibonacci or Lucas numbers
Математичні Студії, 2021 In the present study we investigate all repdigits which are expressed as a difference of two Fibonacci or Lucas numbers. We show that if $F_{n}-F_{m}$ is a repdigit, where $F_{n}$ denotes the $n$-th Fibonacci number, then $(n,m)\in \{(7,3),(9,1),(9,2 ...
P. Ray, K. Bhoi
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Journal of the Indonesian Mathematical Society, 2023 Horadam introduced a generalized sequence of numbers, describing its key features and the special sub-sequences obtained from specific choices of initial parameters. This sequence and its sub-sequences are known as the Horadam, generalized Fibonacci, and generalized Lucas numbers, respectively.
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Exact divisibility by powers of the integers in the Lucas sequence of the first kind
AIMS Mathematics, 2020 Lucas sequence of the first kind is an integer sequence $(U_n)_{n\geq0}$ which depends on parameters $a,b\in\mathbb{Z}$ and is defined by the recurrence relation $U_0=0$, $U_1=1$, and $U_n=aU_{n-1}+bU_{n-2}$ for $n\geq2$. In this article, we obtain exact
Kritkhajohn Onphaeng +1 more
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Oscillatory Nonautonomous Lucas Sequences [PDF]
International Journal of Differential Equations, 2009 The oscillatory behavior of the solutions of the second‐order linear nonautonomous equation x(n + 1) = a(n)x(n) − b(n)x(n − 1), n ∈ ℕ0, where a, b : ℕ0 → ℝ, is studied. Under the assumption that the sequence b(n) dominates somehow a(n), the amplitude of the oscillations and the asymptotic behavior of its solutions are also analized.
Ferreira, José M., Pinelas, Sandra
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On the reciprocal products of generalized Fibonacci sequences
Journal of Inequalities and Applications, 2022 In this paper, we use the properties of error estimation and the analytic method to study the reciprocal products of the bi-periodic Fibonacci sequence, the bi-periodic Lucas sequence, and the mth-order linear recursive sequence.
Tingting Du, Zhengang Wu
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On the intersections of Fibonacci, Pell, and Lucas numbers [PDF]
, 2010 We describe how to compute the intersection of two Lucas sequences of the forms $\{U_n(P,\pm 1) \}_{n=0}^{\infty}$ or $\{V_n(P,\pm 1) \}_{n=0}^{\infty}$ with $P\in\mathbb{Z}$ that includes sequences of Fibonacci, Pell, Lucas, and Lucas-Pell numbers.
Bilu +13 more
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Journal of Number Theory, 2007 Let P and Q be non-zero integers. The Lucas sequence U_n(P,Q) is defined by U_0=0, U_1=1, U_n= P*U_{n-1}-Q*U_{n-2} for n >1. The question of when U_n(P,Q) can be a perfect square has generated interest in the literature. We show that for n=2,...,7, U_n is a square for infinitely many pairs (P,Q) with gcd(P,Q)=1; further, for n=8,...,12, the only non-
Andrew Bremner, Nikos Tzanakis
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Prime divisors of Lucas sequences [PDF]
Acta Arithmetica, 1997 Let \(d>1\) be a squarefree integer and \(\mathbb{K}= \mathbb{Q}(\sqrt{d})\) a real quadratic field. Let \(\varepsilon= a+b\sqrt{d}\) be a fundamental unit in the ring of integers of \(\mathbb{K}\), and \(\overline{\varepsilon}\) its conjugate. The Lucas sequence associated with \(\mathbb{K}\) is the integer sequence defined by \[ X_{\mathbb{K ...
Stevenhagen, P., Moree, P.
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On some new results for the generalised Lucas sequences
Analele Stiintifice ale Universitatii Ovidius Constanta: Seria Matematica, 2021 In this paper we introduce the functions which count the number of generalized Lucas and Pell-Lucas sequence terms not exceeding a given value x and, under certain conditions, we derive exact formulae (Theorems 3 and 4) and establish asymptotic limits ...
Andrica Dorin +2 more
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