• Journal of the Korea Institute of Information Security & Cryptology
• /
• v.4 no.2
• /
• pp.29-38
• /
• 1994

본 논문에서는 오류제어기능을 갖는 정보원 부호화 방식 중 Fibonacci부호와 flag부호에 대하여 알아보고, 부호화와 복호가 간단한 flag 부호에 있어서 비트 삽입 발생에 주목하여 최적의 suffix 패턴을 구한다. 이 최적 suffix 패턴을 사용한 제안부호의 성능을 비트 삽입 발생횟수의 확률적 해석과 컴퓨터 시뮬레이션을 통하여 비교 분석한다. 그 결과, 제안부호의 성능이 평균부호길이면에서 가장 우수한 Fibonacci부호 에 비해 동등이상의 결과를 얻을 수 있음을 보인다.

• Journal of applied mathematics & informatics
• /
• v.40 no.1_2
• /
• pp.243-257
• /
• 2022

In this paper, we present and prove an new theorem on symmetric functions. By using this theorem, we derive some new generating functions of the products of (p, q)-Fibonacci numbers, (p, q)-Lucas numbers, (p, q)-Pell numbers, (p, q)-Pell Lucas numbers, (p, q)-Jacobsthal numbers and (p, q)-Jacobsthal Lucas numbers with Chebyshev polynomials of the first kind.

• Communications of the Korean Mathematical Society
• /
• v.38 no.3
• /
• pp.695-704
• /
• 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.

• Journal of Educational Research in Mathematics
• /
• v.18 no.3
• /
• pp.373-389
• /
• 2008

In this paper, we designed a teaching unit for the learning mathematization of secondary pre-service teachers through exploring the relationship between partition models and generalized fibonacci sequences. We first suggested some problems which guide pre-service teachers to make phainomenon for organizing nooumenon. Pre-service teachers should find patterns from partitions for various partition models by solving the problems and also form formulas front the patterns. A series of these processes organize nooumenon. Futhermore they should relate the formulas to generalized fibonacci sequences. Finding these relationships is a new mathematical material. Based on developing these mathematical materials, pre-service teachers can be experienced mathematising as real practices.

• School Mathematics
• /
• v.11 no.2
• /
• pp.243-260
• /
• 2009

In this paper, we have designed a teaching unit for the learning mathematising of secondary pre-service teachers by exploring generalized fibonacci sequences. First, we have found useful formulas for general terms of generalized fibonacci sequences which are expressed as combinatoric notations. Second, by using these formulas and CAS graphing calculator, we can help secondary pre-service teachers to conjecture and discuss the limit of the sequence given by the rations of two adjacent terms of an m-step fibonacci sequence. These processes can remind secondary pre-service teachers of a series of some mathematical principles.

• Korean Institute of Interior Design Journal
• /
• v.23 no.1
• /
• pp.88-95
• /
• 2014

This study intended arousal of other viewpoints that deal with and understand spaces and shapes, by describing the concept of 'dimensions' into visual patterns. Above all, the core concept of spatial dimensions was defined as 'expandability'. Then, first, the 'golden ratio', 'Fibonacci sequence', and 'fractal theory' were defined as elements of each dimension by stage. Second, a 'unit cell' of one dimension as 'minimum unit particles' was set. Next, Fibonacci sequence was set as an extended concept into two dimensions. Expansion into three dimensions was applied to the concept of 'self-similarity repetition' of 'Fractal'. In 'fractal dimension', the concept of 'regularity of irregularity' was set as a core attribute. Plus, Platonic solids were applied as a background concept of the setting of the 'unit cell' from the viewpoint of 'minimum unit particles'. Third, while 'characteristic patterns' which are shown in the courses of 'expansion' of each dimension were embodied for the visual expression forms of dimensions, expansion forms of dimensions are based on the premise of volume, directional nature, and concept of axes. Expressed shapes of each dimension are shown into visually diverse patterns and unexpected formative aspects, along with the expression of relative blank spaces originated from dualism. On the basis of these results, the 'unit cell' that is set as a concept of theoretical factor can be defined as a minimum factor of a basic algorism caused by other purpose. In here, by applying diverse pattern types, the fact that meaning spaces, shapes, and dimensions can be extracted was suggested.

• Bulletin of the Korean Mathematical Society
• /
• v.52 no.5
• /
• pp.1467-1480
• /
• 2015

Generalized Fibonacci and Lucas sequences ($U_n$) and ($V_n$) are defined by the recurrence relations $U_{n+1}=PU_n+QU_{n-1}$ and $V_{n+1}=PV_n+QV_{n-1}$, $n{\geq}1$, with initial conditions $U_0=0$, $U_1=1$ and $V_0=2$, $V_1=P$. This paper deals with Fibonacci and Lucas numbers of the form $U_n$(P, Q) and $V_n$(P, Q) with the special consideration that $P{\geq}3$ is odd and Q = -1. Under these consideration, we solve the equations $V_n=5kx^2$, $V_n=7kx^2$, $V_n=5kx^2{\pm}1$, and $V_n=7kx^2{\pm}1$ when $k{\mid}P$ with k > 1. Moreover, we solve the equations $V_n=5x^2{\pm}1$ and $V_n=7x^2{\pm}1$.

• Bulletin of the Korean Mathematical Society
• /
• v.51 no.4
• /
• pp.1041-1054
• /
• 2014

Let $P{\geq}3$ be an integer and let ($U_n$) and ($V_n$) denote generalized Fibonacci and Lucas sequences defined by $U_0=0$, $U_1=1$; $V_0= 2$, $V_1=P$, and $U_{n+1}=PU_n-U_{n-1}$, $V_{n+1}=PV_n-V_{n-1}$ for $n{\geq}1$. In this study, when P is odd, we solve the equations $V_n=kx^2$ and $V_n=2kx^2$ with k | P and k > 1. Then, when k | P and k > 1, we solve some other equations such as $U_n=kx^2$, $U_n=2kx^2$, $U_n=3kx^2$, $V_n=kx^2{\mp}1$, $V_n=2kx^2{\mp}1$, and $U_n=kx^2{\mp}1$. Moreover, when P is odd, we solve the equations $V_n=wx^2+1$ and $V_n=wx^2-1$ for w = 2, 3, 6. After that, we solve some Diophantine equations.

• ETRI Journal
• /
• v.40 no.1
• /
• pp.111-121
• /
• 2018

To find the emerging patterns (EPs) in streaming transaction data, the streaming is first divided into some time windows containing a number of transactions. Itemsets are generated from transactions in each window, and then the emergence of itemsets is evaluated between two windows. In the tilted-time windows model (TTWM), it is assumed that people need support data with finer accuracy from the most recent windows, while accepting coarser accuracy from older windows. Therefore, a limited array's elements are used to maintain all support data in a way that condenses old windows by merging them inside one element. The capacity of elements that accommodates the windows inside is modeled using a particular number sequence. However, in a stream, as new data arrives, the current array updating mechanisms lead to many null elements in the array and cause data incompleteness and inaccuracy problems. Two models derived from TTWM, logarithmic TTWM and Fibonacci windows model, also inherit the same problems. This article proposes a novel push-front Fibonacci windows model as a solution, and experiments are conducted to demonstrate its superiority in finding more EPs compared to other models.

• Honam Mathematical Journal
• /
• v.41 no.3
• /
• pp.569-579
• /
• 2019

Cusp (parabolic) points in the extended modular group ${\bar{\Gamma}}$ are basically the images of infinity under the group elements. This implies that the cusp points of ${\bar{\Gamma}}$ are just rational numbers and the set of cusp points is $Q_{\infty}=Q{\cup}\{{\infty}\}$.The Farey graph F is the graph whose set of vertices is $Q_{\infty}$ and whose edges join each pair of Farey neighbours. Each rational number x has an integer continued fraction expansion (ICF) $x=[b_1,{\cdots},b_n]$. We get a path from ${\infty}$ to x in F as $<{\infty},C_1,{\cdots},C_n>$ for each ICF. In this study, we investigate relationships between Fibonacci numbers, Farey graph, extended modular group and ICF. Also, we give a computer program that computes the geodesics, block forms and matrix represantations.