• 한국수학교육학회지시리즈B:이론수학과 교직수학
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• 제31권4호
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• pp.427-437
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• 2024

In this paper, we introduce the Lucas-Padovan quaternions sequence. Initiating the studies based on the Padovan quaternion coefficients in relation to their recurrence, their matrix representation is then defined. We investigate various aspects of these quaternions, including summation formulas and binomial sums.

• 로봇학회논문지
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• 제5권2호
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• pp.85-92
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• 2010

In this paper, we propose a digital image stabilization technique using edge detection and Lucas-Kanade optical flow in order to minimize the motion of the shaken image. The accuracy of motion estimation based on block matching technique depends on the size of search window, which results in long calculation time. Therefore it is not applicable to real-time system. In addition, since the size of vector depends on that of block, it is difficult to estimate the motion which is bigger than the block size. The proposed method extracts the trust region using edge detection, to estimate the motion of some critical points in trust region based on Lucas-Kanade optical flow algorithm. The experimental results show that the proposed method stabilizes the shaking of motion image effectively in real time.

• Kyungpook Mathematical Journal
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• 제63권4호
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• pp.539-550
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• 2023

In this paper, we define the generalized k-balancing numbers {B^(k)_n} and k-Lucas balancing numbers {C^(k)_n} and associated polynomials, where n is of the form sk+r, 0 ≤ r < k. We give several formulas for these new sequences in terms of classic balancing and Lucas balancing numbers and study their properties. Moreover, we give a Binet style formula, Cassini's identity, and binomial sums of these sequences.

• Journal of Applied and Pure Mathematics
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• 제6권5_6호
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• pp.283-299
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• 2024

This article presents a fundamental generalization of the classical Fibonacci sequence. We introduce a general 2^nd order recurrence relation, V_n = pV_n-1 + qV_n-2, q ≠ 0, n ≥ 2, with initial terms V₀(= a), V₁(= b), a, b, p, and q are any non-zero real numbers. We derive an explicit generalized form of the generating function and comprehensive Binet's formula, which comprehend this concept to various sequences that follow similar recurrence relations. Furthermore, we analyze more generalized and specialized cases, uncovering new and existing identities for well-known sequences such as Fibonacci, Lucas, Pell, Pell-Lucas, Goksal Bilgici, and others. Our analysis implicitly reveals identities like Cassini's, Catalan's, d'Ocagne's, and Gelin-Cesàro in the generalized form. Additionally, tabular and graphical representations are provided to illustrate the relationships between the terms of these sequences.

• 대한수학회지
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• 제42권1호
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• pp.135-151
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• 2005

In this paper, we obtain some properties of the sequences $U^{q}_n\;and\;V^{q}_n$ introduced in [6]. We find polynomial representations and formulas of them. For q = 5, $U^{5}_n$ is the Fibonacci sequence $F_n\;and\;V^{5}_n$ is the Lucas sequence $L_n$.

• 대한수학회논문집
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• 제32권3호
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• pp.511-522
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• 2017

We explicitly solve the diophantine equations of the form $$A_{n_1}A_{n_2}{\cdots}A_{n_k}{\pm}1=B^2_m$$, where $(A_n)_{n{\geq}0}$ and $(B_m)_{m{\geq}0}$ are either the Fibonacci sequence or Lucas sequence. This extends the result of D. Marques [9] and L. Szalay [13] concerning a variant of Brocard-Ramanujan equation.

• 대한수학회논문집
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• 제31권3호
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• pp.447-450
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• 2016

Let $F_n$ and $L_n$ be the nth Fibonacci number and Lucas number, respectively. The order of appearance of m in the Fibonacci sequence, denoted by z(m), is the smallest positive integer k such that m divides $F_k$. Marques obtained the formula of $z(L^k_n)$ in some cases. In this article, we obtain the formula of $z(L^k_n)$ for all $n,k{\geq}1$.

• Kyungpook Mathematical Journal
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• 제59권1호
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• pp.23-34
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• 2019

In this paper, we first define generalizations of Pell and Pell-Lucas sequences by the recurrence relations $$p_n=2ap_{n-1}+(b-a^2)p_{n-2}\;and\;q_n=2aq_{n-1}+(b-a^2)q_{n-2}$$ with initial conditions $p_0=0$, $p_1=1$, and $p_0=2$, $p_1=2a$, respectively. We give generating functions and Binet's formulas for these sequences. Also, we obtain some identities of these sequences.

• 대한수학회논문집
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• 제38권3호
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• pp.695-704
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• 2023

Area problems for triangles and polygons whose vertices have Fibonacci numbers on a plane were presented by A. Shriki, O. Liba, and S. Edwards et al. In 2017, V. P. Johnson and C. K. Cook addressed problems of the areas of triangles and polygons whose vertices have various sequences. This paper examines the conditions of triangles and polygons whose vertices have Lucas sequences and presents a formula for their areas.

• 호남수학학술지
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• 제45권4호
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• pp.572-584
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• 2023

In this study, we search for Pell and Pell-Lucas numbers, which are concatenations of two repdigits and find these numbers to be only 12, 29, 70 and 14, 34, 82, respectively. We use Baker's Theory and Baker-Davenport basis reduction method while finding the solutions.