On the representation of k-generalized Fibonacci and Lucas numbers

Öcal, Ahmet Ali; Tuglu, Naim; Altınışık, Ercan

doi:10.1016/j.amc.2004.12.009

Cited by 30 publications

(26 citation statements)

References 11 publications

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“…There are so many studies in the literature that concern about the special number sequences such as Fibonacci, Lucas, Pell, Jacobsthal, Padovan and Perrin (see, for example [1,5,6,8,10,11,13,15], and the references cited therein). Especially, the Fibonacci and Lucas numbers have attracted the attention of mathematicians because of their intrinsic theory and applications.…”

Section: Introductionmentioning

confidence: 99%

A note on the bi-periodic Fibonacci and Lucas matrix sequences

Coskun

Taşkara

2018

Applied Mathematics and Computation

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In this paper, we introduce the bi-periodic Lucas matrix sequence and present some fundamental properties of this generalized matrix sequence. Moreover, we investigate the important relationships between the bi-periodic Fibonacci and Lucas matrix sequences. We express that some behaviours of bi-periodic Lucas numbers also can be obtained by considering properties of this new matrix sequence. Finally, we say that the matrix sequences as Lucas, k-Lucas and Pell-Lucas are special cases of this generalized matrix sequence.

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Section: Introductionmentioning

confidence: 99%

A note on the bi-periodic Fibonacci and Lucas matrix sequences

Coskun

Taşkara

2018

Applied Mathematics and Computation

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“…In [28], the authors obtained some relations between Padovan sequence and permanents of one type of Hessenberg matrix. Kiliç [16] obtained some relations between the Tribonacci sequence and permanents of one type of Hessenberg matrix.Öcal et al [22] studied some determinantal and permanental representations of k-generalized Fibonacci and Lucas numbers. Janjić [14] considered a particular upper Hessenberg matrix and showed its relations with a generalization of the Fibonacci numbers.…”

Section: Introductionmentioning

confidence: 99%

Hessenberg matrices and the generalized Fibonacci-Narayana sequence

Ramírez

2015

Filomat

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In this note, we define the generalized Fibonacci-Narayana sequence {G n (a, b, c)} n∈N. After that, we derive some relations between these sequences, and permanents and determinants of one type of upper Hessenberg matrix. with initial conditions G 0 (a, b, c) = 0, G i (a, b, c) = 1, for i = 1, 2,. .. , c − 1. The constants a and b are nonzero real numbers. We call this sequence generalized Fibonacci-Narayana sequence. Note that, if a = b = 1 and c = 2, the Fibonacci sequence is obtained, and if a = 1 = b and c = 3, the Narayana sequence is obtained [1, 12]. Other particular cases of the sequence {G n (a, b, c)} n∈N are • If a = b = 1, the generalized Fibonacci sequence is obtained [2]. G n = G n−1 + G n−c. • If c = 2, the generalized Fibonacci sequence is obtained. G n = aG n−1 + bG n−2. • If a = k, b = 1 and c = 2, the k-Fibonacci sequence is obtained [10]. F k,n = kF k,n−1 + F k,n−2 .

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“…For example, Minc [23] defined an n × n (0,1)-matrix F (n, k) and showed that the permanents of F (n, k) are equal to the generalized order-k Fibonacci numbers (1.1). The authors [12, 13] defined two (0, 1)-matrices and showed that the permanents of these matrices are the generalized Fibonacci (1.1) and Lucas numbers.Öcal et al [24] gave some determinantal and permanental representations of k-generalized Fibonacci and Lucas numbers and obtained Binet's formula for these sequences. Kılıç and Stakhov [9] gave permanent representation of Fibonacci and Lucas p-numbers.…”

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confidence: 99%

Determinantal and permanental representations of Fibonacci type numbers and polynomials

Kaygisiz¹,

Sahin²

2016

Rocky Mountain J. Math.

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In this paper, we compute terms of the matrix A ∞ (k) , which contains Fibonacci type numbers and polynomials, with the help of determinants and permanents of various Hessenberg matrices. In addition, we show that determinants of these Hessenberg matrices can be obtained by using combinations. The results that we obtain are important, since the matrix A ∞ (k) is a general form of Fibonacci type numbers and polynomials, such as k sequences of the generalized order-k Fibonacci and Pell numbers, generalized bivariate Fibonacci p-polynomials, bivariate Fibonacci and Pell p-polynomials, second kind Chebyshev polynomials and bivariate Jacobsthal polynomials, etc. 2010 AMS Mathematics subject classification. Primary 11B37, Secondary 15A15, 15A51.Keywords and phrases. Matrix A ∞ (k) , generalized Fibonacci polynomials, Hessenberg matrix.

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On the representation of k-generalized Fibonacci and Lucas numbers

Cited by 30 publications

References 11 publications

A note on the bi-periodic Fibonacci and Lucas matrix sequences

A note on the bi-periodic Fibonacci and Lucas matrix sequences

Hessenberg matrices and the generalized Fibonacci-Narayana sequence

Determinantal and permanental representations of Fibonacci type numbers and polynomials

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