• 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.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.

• 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.

• 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.

• 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.

• Kyungpook Mathematical Journal
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• v.55 no.2
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• pp.355-372
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• 2015

In the present paper, by using the Fibonacci difference matrix, we introduce the almost convergent sequence space $\hat{c}^f$. Also, we show that the spaces $\hat{c}^f$and $\hat{c}$ are linearly isomorphic. Further, we determine the ${\beta}$-dual of the space $\hat{c}^f$ and characterize some matrix classses on this space. Finally, Fibonacci core of a complex-valued sequence has been introduced, and we prove some inclusion theorems related to this new type of core.

• Journal for History of Mathematics
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• v.13 no.1
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• pp.63-76
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• 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 applied mathematics & informatics
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• v.19 no.1_2
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• pp.231-240
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• 2005

We examine the Fibonacci length of certain centro-polyhedral groups and show that in some cases the lengths depend on tribonacci sequences. Further we obtain specific examples of infinite families of three-generator groups with constant, linear and (3-step) Wall number dependent Fibonacci lengths.

• 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.

• 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 {$q_n$}.