• School Mathematics
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• v.11 no.2
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• pp.243-260
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• 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 for History of Mathematics
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• v.21 no.4
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• pp.87-104
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• 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 $\frac{1+\sqrt5}2$.

• Journal of Educational Research in Mathematics
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• v.18 no.3
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• pp.373-389
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• 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
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• v.12 no.4
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• pp.619-638
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• 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.

• The Journal of the Institute of Internet, Broadcasting and Communication
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• v.16 no.6
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• pp.319-327
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• 2016

In this paper we investigate that most of plants and animals have the symmetric property, such as a tree or a sheep's horn. In addition, the human body is also symmetric and contains the DNA. We can see the logarithm helices in Fibonacci series and animals, and helices of plants. The sunflower has a shape of circle. A circle is circular symmetric because the shapes are same when it is shifted on the center. Einstein's spatial relativity is the relation of time and space conversion by the symmetrically generalization of time and space conversion over the spacial. The left and right helices of plants and animals are the symmetric and have element-wise inverse relationships each other. The weight of center weight Hadamard matrix is 2 and is same as the base 2 of natural logarithm. The helix matrices are symmetric and have element-wise inverses.