• Journal for History of Mathematics
• /
• v.21 no.4
• /
• pp.87-104
• /
• 2008

In this paper we investigate several properties and characteristics of the generalized Fibonacci sequence $\{g_n\}$={a, b, a+b, a+2b, 2a+3b, 3a+5b,...}. This concept is a generalization of the famous Fibonacci sequence. In particular we find the identities of sums and the nth term $g_n$ in detail. Also we find the generalizations of the Catalan's identity and A. Tagiuri's identity about the Fibonacci sequence, and investigate the relation between $g_n$ and Pascal's triangle, and how fast $g_n$ increases. Furthermore, we show that $g_n$ and $g_{n+1}$ are relatively prime if a b are relatively prime, and that the sequence $\{\frac{g_{n+1}}{g_n}\}$ of the ratios of consecutive terms converges to the golden ratio $\frac{1+\sqrt5}2$.

• The Korean Journal of Applied Statistics
• /
• v.22 no.5
• /
• pp.1103-1113
• /
• 2009

We know that the first digits sequence of fibonacci numbers obey Benford's law. For the sequence in which the first two numbers are the arbitrary integers and the recurrence relation $a_{n+2}=a_{n+1}+a_n$ is satisfied, we can find that the first digits sequence of this sequence obey Benford's law. Also, we can find the stucture of the first digits sequence of this sequence with the exploratory data analysis tools.

• 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.

• 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.12 no.4
• /
• pp.619-638
• /
• 2010

In this paper, we listed and reviewed some properties on polygonal numbers, pyramidal numbers and Pascal's triangle, and Fibonacci sequence. We discussed that the properties of gnomonic numbers, polygonal numbers and pyramidal numbers are explained integratively by introducing the generalized k-dimensional pyramidal numbers. And we also discussed that the properties of those numbers and relationships among generalized k-dimensional pyramidal numbers, Pascal's triangle and Fibonacci sequence are suitable for teaching and learning of mathematical reasoning and connections.

• Journal for History of Mathematics
• /
• v.13 no.1
• /
• pp.63-76
• /
• 2000

In this paper we survey the main properties of a Fibonacci sequence, and find out examples of Fibonacci sequence in our nature and daily life.

• Journal of the Institute of Convergence Signal Processing
• /
• v.19 no.2
• /
• pp.42-46
• /
• 2018

Adaptive signal processing is quite important in various signal and communication environments. In adaptive signal processing methods since the least mean square(LMS) algorithm is simple and robust, it is used everywhere. As the step is varied in the variable step(VS) LMS algorithm, the fast convergence speed and the small excess mean square error can be obtained. Various variable step LMS algorithms are researched for better performances. But in some of variable step LMS algorithms the computational complexity is quite large for better performances. The fixed step LMS algorithm with a low computational complexity merit and the variable step LMS algorithm with a fast convergence merit are combined in the proposed sporadic step algorithm. As the step is sporadically updated, the performances of the variable step LMS algorithm can be maintained in the low update rate using Fibonacci sequence. The performances of the proposed variable step LMS algorithm are proved in the adaptive equalizer.

• Communications of Mathematical Education
• /
• v.27 no.4
• /
• pp.357-367
• /
• 2013

It is important to make students do research for oneself. But the practice of inquiry activity is not easy in the mathematics education field. Intellectual curiosities of students are unpredictable. It is important to meet intellectual curiosities of students. We could get a sequence in the process solving a problem. This sequence was expressed in a form of the recurrence relation $a_n=a_{n-1}+a_{n-3}$ ($n{\geq}4$), $a_1=a_2=a_3=1$. We tried to look for the general terms of this sequence. This sequence is similar to Fibonacci sequence, but the process finding the general terms is never similar to Fibonacci sequence. We can get two general terms expressed in different form after our a great deal of effort. We hope that this study will give the spot of education energy.

• 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.

• Journal of the Korean housing association
• /
• v.15 no.5
• /
• pp.69-76
• /
• 2004

비트루비우스(Vitruvius)에서 벤츄리(Venturi)에 이르기까지 많은 건축사가(建榮史家) 또는 건축가(建築家)들이 건축원리에 대한 각자의 규범적 태도를 글로써 밝혀왔다. 이러한 규범적 내용 중 비례이론은 서양건축에 있어 건축미(建築美) 비밀로 간주되어, 조화로운 건축물이 꼭 갖추어야 할 덕목중의 하나로 인식되었다. 현대건축에서 비교적 근대 비례시스템이라 불릴 수 있는 것으로 르 꼬르뷔지에의 모될로르와 반 데어 란의 플라스틱 넘버를 들 수 있다. 모될로르는 현재까지 아주 제한된 성공밖에 거두지 못했는데, 이는 피보나치 수열에 기초한 수치들이 커졌을 때 단위의 배수관계가 거의 형성되지 않는다는 사실에 기인한다. 반면에 플라스틱 넘버는 모될로르의 결점들을 보완할 수 있는 매력적인 수열을 가지고 있다. 이에 본 연구는 반 데어 란의 비례시스템 분석을 통하여 유도된 수열규칙이 건축디자인과 관련되어 직접적으로 적용되어 질 수 있는 CAAD시스템을 제안한다. 본 시스템의 초점은 반 데어 란의 비례이론이 어떻게 컴퓨터 언어로 변환 및 CAAD시스템에 적용되어, 실질적인 건축 실무행위에 있어 컴퓨터가 디자인 도우미로서 역할을 수행할 수 있겠는가하는 것이다. 연구의 결과, 사용자는 이러한 시스템을 사용함으로써 반 데어 란의 비례시스템을 자신의 디자인에 손쉽게 적용할 수 있으며, 이는 복잡한 치수관계로 구성된 비례시스템의 건축실무 활용으로 발전되어질 수 있을 것으로 사료된다.