• 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.

• Honam Mathematical Journal
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• v.34 no.4
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• pp.533-548
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• 2012

We investigate relations of polynomials $x^n={\Sigma}^{n-1}_{i=0}x^i$ and recurrence sequences. We consider certain variations of the Fibonacci sequence and investigate explicit ways to compute the general nth term.

• Journal of the Korean Mathematical Society
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• v.40 no.6
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• pp.921-932
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• 2003

A number of variants of Hofstadter's original sequence have been investigated. Here we investigate a collection of similarly defined such sequences which give rise to intriguingly different results.

• Communications of the Korean Mathematical Society
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• v.38 no.2
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• pp.341-354
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• 2023

In this article, an alternating Fibonacci sequence is defined from a second-order linear homogeneous recurrence relation with constant coefficients. Then, the determinant, inverse, and eigenvalues of the circulant matrices with entries in the first row having the formation of the sequence are formulated explicitly in a simple way. In this study, the method for deriving the formulation of the determinant and inverse is simply using traditional elementary row or column operations. For the eigenvalues, the known formulation from the case of general circulant matrices is simplified by considering the specialty of the sequence and using cyclic group properties. We also propose algorithms for the formulation to show how efficient the computations are.

• Kyungpook Mathematical Journal
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• v.49 no.3
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• pp.483-492
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• 2009

In this paper we give formulae for sums of products of two Horadam type generalized Fibonacci numbers with the same recurrence equation and with possibly different initial conditions. Analogous improved alternating sums are also studied as well as various derived sums when terms are multiplied either by binomial coefficients or by members of the sequence of natural numbers. These formulae are related to the recent work of Belbachir and Bencherif, $\v{C}$erin and $\v{C}$erin and Gianella.

• Communications of the Korean Mathematical Society
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• v.37 no.4
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• pp.977-988
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• 2022

The Padovan sequence is the third-order linear recurrence (𝓟_n)_n≥0 defined by 𝓟_n = 𝓟_n-2 + 𝓟_n-3 for all n ≥ 3 with initial conditions 𝓟₀ = 0 and 𝓟₁ = 𝓟₂ = 1. In this paper, we investigate a generalization of the Padovan sequence called the k-generalized Padovan sequence which is generated by a linear recurrence sequence of order k ≥ 3. We present recurrence relations, the generalized Binet formula and different arithmetic properties for the above family of sequences.

• 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.

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

• The Pure and Applied Mathematics
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• v.20 no.3
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• pp.207-221
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• 2013

In this work we study the tribonacci numbers. We find a tribonacci triangle which is an analog of Pascal triangle. We also investigate an efficient method to compute any $n$th tribonacci numbers by matrix method, and find periods of the sequence by taking modular tribonacci number.

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