• Title/Summary/Keyword: generalized Fibonacci sequence.

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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.

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.

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 {$q_n$}.

A Study on Teaching Material for Enhancing Mathematical Reasoning and Connections - Figurate numbers, Pascal's triangle, Fibonacci sequence - (수학적 추론과 연결성의 교수.학습을 위한 소재 연구 -도형수, 파스칼 삼각형, 피보나치 수열을 중심으로-)

  • Son, Hong-Chan
    • 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.

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A Study on Generalized Fibonacci Sequence (피보나치 수열의 일반화에 관한 고찰)

  • Yang, Young-Oh;Kim, Tae-Ho
    • Journal for History of Mathematics
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    • v.21 no.4
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    • pp.87-104
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    • 2008
  • In this paper we investigate several properties and characteristics of the generalized Fibonacci sequence $\{g_n\}$={a, b, a+b, a+2b, 2a+3b, 3a+5b,...}. This concept is a generalization of the famous Fibonacci sequence. In particular we find the identities of sums and the nth term $g_n$ in detail. Also we find the generalizations of the Catalan's identity and A. Tagiuri's identity about the Fibonacci sequence, and investigate the relation between $g_n$ and Pascal's triangle, and how fast $g_n$ increases. Furthermore, we show that $g_n$ and $g_{n+1}$ are relatively prime if a b are relatively prime, and that the sequence $\{\frac{g_{n+1}}{g_n}\}$ of the ratios of consecutive terms converges to the golden ratio $\frac{1+\sqrt5}2$.

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A COMPLETENESS ON GENERALIZED FIBONACCI SEQUENCES

  • Lee, Gwang-Yeon
    • 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.

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ON CONDITIONALLY DEFINED FIBONACCI AND LUCAS SEQUENCES AND PERIODICITY

  • Irby, Skylyn;Spiroff, Sandra
    • 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.

GENERALIZED PADOVAN SEQUENCES

  • Bravo, Jhon J.;Herrera, Jose L.
    • 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 (𝓟n)n≥0 defined by 𝓟n = 𝓟n-2 + 𝓟n-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.