• Title, Summary, Keyword: Fibonacci sequence

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A NOTE ON THE MODIFIED k-FIBONACCI-LIKE SEQUENCE

  • Kwon, Youngwoo
    • 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.

Some Properties of Fibonacci Numbers, Generalized Fibonacci Numbers and Generalized Fibonacci Polynomial Sequences

  • Laugier, Alexandre;Saikia, Manjil P.
    • 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.

SUM FORMULAE OF GENERALIZED FIBONACCI AND LUCAS NUMBERS

  • Cerin, Zvonko;Bitim, Bahar Demirturk;Keskin, Refik
    • 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.

CATALAN TRANSFORM OF THE κ-FIBONACCI SEQUENCE

  • Falcon, Sergio
    • 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.

(±1)-INVARIANT SEQUENCES AND TRUNCATED FIBONACCI SEQUENCES OF THE SECOND KIND

  • CHOI GYOUNG-SIK;HWANG SUK-GEUN;KIM IK-PYO
    • Journal of the Korean Mathematical Society
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    • v.42 no.4
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    • pp.761-771
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    • 2005
  • In this paper we present another characterization of (${\pm}1$)-invariant sequences. We also introduce truncated Fibonacci and Lucas sequences of the second kind and show that a sequence $x\;{\in}\;R^{\infty}$ is (-1)-invariant(l-invariant resp.) if and only if $D[_x^0]$ is perpendicular to every truncated Fibonacci(truncated Lucas resp.) sequence of the second kind where $$D=diag((-1)^0,\; (-1)^1,\;(-1)^2,{\ldots})$$.

On the Fibonacci Almost Convergent Sequence Space and Fibonacci Core

  • DEMIRIZ, SERKAN;KARA, EMRAH EVREN;BASARIR, METIN
    • 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.

피보나치 수열에 관한 고찰

  • 양영오
    • 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.

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COMPLEX FACTORIZATIONS OF THE GENERALIZED FIBONACCI SEQUENCES {qn}

  • JUN, SANG PYO
    • 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 {$q_n$}.