• Honam Mathematical Journal
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• v.40 no.1
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• pp.199-210
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• 2018

In this paper we obtain some formulae for several sums of generalized Fibonacci numbers $U_n$ and generalized Lucas numbers $V_n$ and their dual forms $G_n$ and $H_n$ by using extensions of an interesting identity by A. R. Amini for Fibonacci numbers to these four kinds of generalizations and their first and second derivatives.

• Bulletin of the Korean Mathematical Society
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• v.56 no.2
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• pp.535-547
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• 2019

In this paper, by using the lower bound of linear forms in logarithms of Matveev and the theory of continued fractions by means of a variation of a result of Dujella and $Peth{\ddot{o}}$, we find all generalized Fibonacci numbers which are Pell numbers. This paper continues a previous work that searched for Pell numbers in the Fibonacci sequence.

• 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.

• Communications of the Korean Mathematical Society
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• v.29 no.3
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• pp.387-399
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• 2014

The ratios of any two Fibonacci numbers are expressed by means of semisimple continued fraction.

• Journal of the Korean Mathematical Society
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• v.48 no.1
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• pp.33-48
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• 2011

The notion of the generalized Fibonacci matrix $\mathcal{F}_n^{(a,b,s)}$ of type s, whose nonzero elements are generalized Fibonacci numbers, is introduced in the paper [23]. Regular case s = 0 is investigated in [23]. In the present article we consider singular case s = -1. Pseudoinverse of the generalized Fibonacci matrix $\mathcal{F}_n^{(a,b,-1)}$ is derived. Correlations between the matrix $\mathcal{F}_n^{(a,b,-1)}$ and the Pascal matrices are considered. Some combinatorial identities involving generalized Fibonacci numbers are derived. A class of test matrices for computing the Moore-Penrose inverse is presented in the last section.

• Journal of applied mathematics & informatics
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• v.20 no.1_2
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• pp.171-180
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• 2006

Two infinite classes of special finite groups considered (The group G is special, if G' and Z(G) coincide). Using certain sequences of numbers we give explicit formulas for the Fibonacci lenghts of these classes which involve the well-known Wall numbers k(n).

• Honam Mathematical Journal
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• v.42 no.2
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• pp.319-330
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• 2020

We introduce a new subclass of q-bi-univalent functions of complex order related to shell-like curves connected with the Fibonacci numbers. We obtain the coefficient estimates and Fekete-Szegö inequalities for the functions belonging to this class. Relevant connections with various other known classes have been illustrated.

• Communications of the Korean Mathematical Society
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• v.28 no.4
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• pp.827-832
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• 2013

In this paper we apply the Catalan transform to the ${\kappa}$-Fibonacci sequence finding different integer sequences, some of which are indexed in OEIS and others not. After we apply the Hankel transform to the Catalan transform of the ${\kappa}$-Fibonacci sequence and obtain an unusual property.

• Communications of the Korean Mathematical Society
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• v.31 no.1
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• pp.1-16
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• 2016

The Fibonacci sequence is a sequence of numbers that has been studied for hundreds of years. In this paper, we introduce the modified k-Fibonacci-like sequence and prove Binet's formula for this sequence and then use it to introduce and prove the Catalan, Cassini, and d'Ocagne identities for the modified k-Fibonacci-like sequence. Also, the ordinary generating function of this sequence is stated.

• Honam Mathematical Journal
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• v.40 no.1
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• pp.139-153
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• 2018

Let $P{\geq}3$ be an integer and let ($U_n$) denote generalized Fibonacci sequence defined by $U_0=0$, $U_1=1$ and $U_{n+1}=PU_n-U_{n-1}$ for $n{\geq}1$. In this study, when P is odd, we solve the equation $U_n=11x^2+1$. We show that only $U_1$ and $U_2$ may be of the form $11x^2+1$.