Lucas number ≈ Lucas number
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On Jacobsthal and Jacobsthal-Lucas Hybrid Numbers Open
The hybrid numbers are generalization of complex, hyperbolic and dual numbers. In this paper we consider special kinds of hybrid numbers, namely the Jacobsthal and the Jacobsthal-Lucas hybrid numbers and we give some their properties.
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Some Combinatorial Properties of the k-Fibonacci and the k-Lucas Quaternions Open
In this paper, we define the k-Fibonacci and the k-Lucas quaternions. We investigate the generating functions and Binet formulas for these quaternions. In addition, we derive some sums formulas and identities such as Cassini’s identity.
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Novel Fibonacci and non-Fibonacci structure in the sunflower: results of a citizen science experiment Open
This citizen science study evaluates the occurrence of Fibonacci structure in the spirals of sunflower ( Helianthus annuus ) seedheads. This phenomenon has competing biomathematical explanations, and our core premise is that observation of…
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On quaternions with generalized Fibonacci and Lucas number components Open
In this paper, we give the exponential generating functions for the generalized Fibonacci and generalized Lucas quaternions, respectively. Moreover, we give some new formulas for binomial sums of these quaternions by using their Binet form…
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Pell and Pell-Lucas numbers with only one distinct digit Open
In this paper, we show that there are no Pell or Pell-Lucas numbers larger
\nthan 10 with only one distinct digit.
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Novel Results for Two Generalized Classes of Fibonacci and Lucas Polynomials and Their Uses in the Reduction of Some Radicals Open
The goal of this study is to develop some new connection formulae between two generalized classes of Fibonacci and Lucas polynomials. Hypergeometric functions of the kind 2F1(z) are included in all connection coefficients for a specific z.…
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The upper bound estimation on the spectral norm of r-circulant matrices with the Fibonacci and Lucas numbers Open
Let us define $A=\operatorname{Circ}_{r}(a_{0},a_{1},\ldots,a_{n-1})$ to be a $n\times n$ r-circulant matrix. The entries in the first row of $A=\operatorname{Circ}_{r}(a_{0},a_{1},\ldots,a_{n-1})$ are $a_{i}=F_{i}$ , or $a_{i}=L_{i}$ , or…
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A Study on Dual Hyperbolic Fibonacci and Lucas Numbers Open
In this study, the dual-hyperbolic Fibonacci and dual-hyperbolic Lucas numbers are introduced. Then, the fundamental identities are proven for these numbers. Additionally, we give the identities regarding negadual-hyperbolic Fibonacci and …
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Some Identities Involving Fibonacci Polynomials and Fibonacci Numbers Open
The aim of this paper is to research the structural properties of the Fibonacci polynomials and Fibonacci numbers and obtain some identities. To achieve this purpose, we first introduce a new second-order nonlinear recursive sequence. Then…
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On the k-Mersenne–Lucas numbers Open
In this paper, we will introduce a new definition of k-Mersenne–Lucas numbers and investigate some properties. Then, we obtain some identities and established connection formulas between k-Mersenne–Lucas numbers and k-Mersenne numbers thro…
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Fibonacci Numbers with a Prescribed Block of Digits Open
In this paper, we prove that F 22 = 17711 is the largest Fibonacci number whose decimal expansion is of the form a b … b c … c . The proof uses lower bounds for linear forms in three logarithms of algebraic numbers and some tools from Diop…
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Mersenne Lucas numbers and complete homogeneous symmetric functions Open
In this paper, we first introduce new definition of Mersenne Lucas numbers sequence as, for n 2, m n = 3m n-1 -2m n-2 with the initial conditions m 0 = 2 and m 1 = 3. Considering this sequence, we give Binet's formula, generating function …
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On Generalized Fibonacci Polynomials: Horadam Polynomials Open
In this paper, we investigate the generalized Fibonacci (Horadam) polynomials and we deal with, in detail, two special cases which we call them $(r,s)$-Fibonacci and $(r,s)$-Lucas polynomials. We present Binet's formulas, generating functi…
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A Note On Bicomplex Fibonacci and Lucas Numbers Open
In this study, we define a new type of Fibonacci and Lucas num- bers which are called bicomplex Fibonacci and bicomplex Lucas numbers. We obtain the well-known properties e.g. Docagnes, Cassini, Catalan for these new types. We also give th…
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On complex Leonardo numbers Open
In this study, we introduce the complex Leonardo numbers and give some of their properties including Binet formula, generating function, Cassini and d’Ocagne’s identities. Also, we calculate summation formulas for complex Leonardo numbers …
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The $ p $-Frobenius and $ p $-Sylvester numbers for Fibonacci and Lucas triplets Open
In this paper we study a certain kind of generalized linear Diophantine problem of Frobenius. Let $ a_1, a_2, \dots, a_l $ be positive integers such that their greatest common divisor is one. For a nonnegative integer $ p $, denote the $ p…
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Mersenne-Lucas hybrid numbers Open
We introduce Mersenne-Lucas hybrid numbers. We give the Binet formula, the generating function, the sum, the character, the norm and the vector representation of these numbers. We find some relations among Mersenne-Lucas hybrid numbers, Ja…
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2-Fibonacci polynomials in the family of Fibonacci numbers Open
In the present study, we define new 2-Fibonacci polynomials by using terms of a new family of Fibonacci numbers given in [4].We show that there is a relationship between the coefficient of the 2-Fibonacci polynomials and Pascal's triangle.…
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Identities of the Chebyshev Polynomials, the Inverse of a Triangular Matrix, and Identities of the Catalan Numbers Open
In the paper, the authors establish two identities to express the generating function of the Chebyshev polynomials of the second kind and its higher order derivatives in terms of the generating function and its derivatives each other, dedu…
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New Banach Sequence Spaces That Is Defined By The Aid Of Lucas Numbers Open
In this work, we establish a new matrix by using Lucas numbers and define a new sequence space.Besides, we give some inclusion relations and investigate the geometrical properties such as Banach-Saks type , weak fixed point property for th…
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Some properties of Fibonacci numbers, Fibonacci octonions, and generalized Fibonacci-Lucas octonions Open
In this paper we determine some properties of Fibonacci octonions.Also, we introduce the generalized Fibonacci-Lucas octonions and we investigate some properties of these elements.
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On the <i>k</i> -Lucas Numbers and Lucas Polynomials Open
In this paper, we introduce an operator in order to derive some new symmetric properties of -Lucas numbers and Lucas polynomials. By making use of the operator defined in this paper, we give some new generating functions for -Lucas numbers…
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The reciprocal sums of even and odd terms in the Fibonacci sequence Open
In this paper, we investigate the reciprocal sums of even and odd terms in the Fibonacci sequence, and we obtain four interesting families of identities which give the partial finite sums of the even-indexed (resp., odd-indexed) reciprocal…
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THE QUOTIENT SET OF -GENERALISED FIBONACCI NUMBERS IS DENSE IN Open
The quotient set of $A\subseteq \mathbb{N}$ is defined as $R(A):=\{a/b:a,b\in A,b\neq 0\}$ . Using algebraic number theory in $\mathbb{Q}(\sqrt{5})$ , Garcia and Luca [‘Quotients of Fibonacci numbers’, Amer. Math. Monthly , to appear] prov…
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On numbers n dividing the nth term of a Lucas sequence Open
We prove that if [Formula: see text] is a Lucas sequence satisfying some mild hypotheses, then the number of positive integers [Formula: see text] does not exceed [Formula: see text] and such that [Formula: see text] divides [Formula: see …
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On Terms of Generalized Fibonacci Sequences which are Powers of their Indexes Open
The k-generalized Fibonacci sequence ( F n ( k ) ) n (sometimes also called k-bonacci or k-step Fibonacci sequence), with k ≥ 2 , is defined by the values 0 , 0 , … , 0 , 1 of starting k its terms and such way that each term afterwards is …
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Hyperbolic Fibonacci Sequence Open
In this paper, we investigate the hyperbolic Fibonacci sequence and the hyperbolic Fibonacci numbers. Furthermore, we give recurrence relations, the golden ratio and Binet's formula for the hyperbolic Fibonacci sequence and Lorentzian inne…
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Applied Mathematics & Information Sciences Open
This paper addresses the generalized Euler polynomial matrix E (α) (x) and the Euler matrix E . Taking into account some properties of Euler polynomials and numbers, we deduce product formulae for E (α) (x) and define the inverse matrix of…
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The Frobenius Number for Jacobsthal Triples Associated with Number of Solutions Open
In this paper, we find a formula for the largest integer (p-Frobenius number) such that a linear equation of non-negative integer coefficients composed of a Jacobsthal triplet has at most p representations. For p=0, the problem is reduced …
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Some results on the q-analogues of the incomplete Fibonacci and Lucas Polynomials Open
In the present paper, we introduce new families of the q-Fibonacci and q-Lucas polynomials, which are represented here as the incomplete q-Fibonacci polynomials F k n .x;s; q/ and the incomplete q-Lucas polynomials L k n .x;s; q/, respecti…