• Kyungpook Mathematical Journal
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• v.57 no.1
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• pp.1-84
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• 2017

In this paper we study the Fibonacci numbers and derive some interesting properties and recurrence relations. We prove some charecterizations for $F_p$, where p is a prime of a certain type. We also define period of a Fibonacci sequence modulo an integer, m and derive certain interesting properties related to them. Afterwards, we derive some new properties of a class of generalized Fibonacci numbers. In the last part of the paper we introduce some generalized Fibonacci polynomial sequences and we derive some results related to them.

• Korean Journal of Mathematics
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• v.23 no.3
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• pp.371-377
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• 2015

In this note, we consider a generalized Fibonacci sequence {$q_n$}. Then give a connection between the sequence {$q_n$} and the Chebyshev polynomials of the second kind $U_n(x)$. With the aid of factorization of Chebyshev polynomials of the second kind $U_n(x)$, we derive the complex factorizations of the sequence {$q_n$}.

• Korean Journal of Mathematics
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• v.22 no.3
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• pp.443-454
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• 2014

In this paper, we give linearization of generalized Fi-bonacci sequences {$g_n$} and {$q_n$}, respectively, defined by Eq.(5) and Eq.(6) below and use this result to give the matrix form of the nth power of a companion matrix of {$g_n$} and {$q_n$}, respectively. Then we re-prove the Cassini's identity for {$g_n$} and {$q_n$}, respectively.

• Korean Journal of Mathematics
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• v.24 no.4
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• pp.681-691
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• 2016

In this note, we consider a generalized Fibonacci sequence {$q_n$}. We give a generating matrix for {$q_n$}. With the aid of this matrix, we derive and re-prove some properties involving terms of this sequence.

• Bulletin of the Korean Mathematical Society
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• v.57 no.4
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• pp.1033-1048
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• 2020

We synthesize the recent work done on conditionally defined Lucas and Fibonacci numbers, tying together various definitions and results generalizing the linear recurrence relation. Allowing for any initial conditions, we determine the generating function and a Binet-like formula for the general sequence, in both the positive and negative directions, as well as relations among various sequence pairs. We also determine conditions for periodicity of these sequences and graph some recurrent figures in Python.

• 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.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.

• Bulletin of the Korean Mathematical Society
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• v.32 no.2
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• pp.239-249
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• 1995

Let $V = (v_1, v_2, \cdots)$ be a sequence of positive integers arranged in nondecreasing order. We define V to be complete if every positive integer n is the sum of some subsequence of V, that is, $$ (1.1) n = \sum_{i=1}^{\infty} a_i v_i where a_i = 0 or 1.

• 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.

• Bulletin of the Korean Mathematical Society
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• v.52 no.5
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• pp.1467-1480
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• 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 $V_n=7x^2{\pm}1$.