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Journal ArticleDOI

On the k-Mersenne–Lucas numbers

01 Mar 2021-Vol. 27, Iss: 1, pp 7-13
TL;DR: In this article, a new definition of k-Mersenne-Lucas numbers was introduced and properties of these numbers were investigated, and some identities and established connection formulas between them through the use of Binet's formula.
Abstract: 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 through the use of Binet’s formula.

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Citations
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Journal ArticleDOI
TL;DR: In this article, a new concept of bivariate Mersenne Lucas polynomials is introduced and the recurrence relation of them is given, and Binet's formula, generating function, Catalan's identity and Cassini's identity for this type of polynomial are obtained.
Abstract: The main aim of this paper is to introduce new concept of bivariate Mersenne Lucas polynomials { m n ( x , y ) } n = 0 ∞ , we first give the recurrence relation of them. We then obtain Binet’s formula, generating function, Catalan’s identity and Cassini’s identity for this type of polynomials. After that, we give the symmetric function, explicit formula and d’Ocagne’s identity of bivariate Mersenne and bivariate Mersenne Lucas polynomials. By using the Binet’s formula we obtain some well-known identities of these bivariate polynomials. Also, some summation formulas of bivariate Mersenne and bivariate Mersenne Lucas polynomials are investigated.

4 citations

Journal ArticleDOI
TL;DR: In this article , the Mersenne hybrinomial quaternions were introduced and some properties of these polynomials were derived, such as Binet formulas, Catalan, Cassini, d'Ocagne identity and generating and exponential generating function.
Abstract: In this paper, we introduce Mersenne and Mersenne-Lucas hybrinomial quaternions and present some of their properties. Some identities are derived for these polynomials. Furthermore, we give the Binet formulas, Catalan, Cassini, d’Ocagne identity and generating and exponential generating function of these hybrinomial quaternions.

1 citations

25 Jul 2022
TL;DR: The closed form formulae for these octonions are given and some well-known identities like Cassini's identity, d’Ocagne’s identity, Catalan identity, Vajda's identity and generating functions of them are obtained.
Abstract: This paper aims to introduce the k -Mersenne and k -Mersenne-Lucas octonions. We give the closed form formulae for these octonions and obtain some well-known identities like Cassini’s identity, d’Ocagne’s identity, Catalan identity, Vajda’s identity and generating functions of them. As a consequence k = 1 yields all the above properties for Mersenne and Mersenne-Lucas octonions.

1 citations

16 Jul 2023
TL;DR: In this article , a higher order Mersenne sequence is introduced and studied, which is analogous to the higher order Fibonacci numbers and closely associated with the Mersenzenne numbers.
Abstract: In this article, we introduce and study a new integer sequence referred to as the higher order Mersenne sequence. The proposed sequence is analogous to the higher order Fibonacci numbers and closely associated with the Mersenne numbers. Here, we discuss various algebraic properties such as Binet's formula, Catalan's identity, d'Ocagne's identity, generating functions, finite and binomial sums, etc. of this new sequence, and some inter-relations with Mersenne and Jacobsthal numbers. Moreover, we study the sequence generated from the binomial transforms of the higher order Mersenne numbers and present the recurrence relation and algebraic properties of them. Lastly, we give matrix generators and tridiagonal matrix representation for higher order Mersenne numbers.
References
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Book
01 Jan 2001
TL;DR: The first 100 Lucas Numbers and their prime factorizations were given in this article, where they were shown to be a special case of the first 100 Fibonacci Numbers and Lucas Polynomials.
Abstract: Preface. List of Symbols. Leonardo Fibonacci. The Rabbit Problem. Fibonacci Numbers in Nature. Fibonacci Numbers: Additional Occurrances. Fibonacci and Lucas Identities. Geometric Paradoxes. Generalized Fibonacci Numbers. Additional Fibonacci and Lucas Formulas. The Euclidean Algorithm. Solving Recurrence Relations. Completeness Theorems. Pascal's Triangle. Pascal-Like Triangles. Additional Pascal-Like Triangles. Hosoya's Triangle. Divisibility Properties. Generalized Fibonacci Numbers Revisited. Generating Functions. Generating Functions Revisited. The Golden Ratio. The Golden Ratio Revisited. Golden Triangles. Golden Rectangles. Fibonacci Geometry. Regular Pentagons. The Golden Ellipse and Hyperbola. Continued Fractions. Weighted Fibonacci and Lucas Sums. Weighted Fibonacci and Lucas Sums Revisited. The Knapsack Problem. Fibonacci Magic Squares. Fibonacci Matrices. Fibonacci Determinants. Fibonacci and Lucas Congruences. Fibonacci and Lucas Periodicity. Fibonacci and Lucas Series. Fibonacci Polynomials. Lucas Polynomials. Jacobsthal Polynomials. Zeros of Fibonacci and Lucas Polynomials. Morgan-Voyce Polynomials. Fibonometry. Fibonacci and Lucas Subscripts. Gaussian Fibonacci and Lucas Numbers. Analytic Extensions. Tribonacci Numbers. Tribonacci Polynomials. Appendix 1: Fundamentals. Appendix 2: The First 100 Fibonacci and Lucas Numbers. Appendix 3: The First 100 Fibonacci Numbers and Their Prime Factorizations. Appendix 4: The First 100 Lucas Numbers and Their Prime Factorizations. References. Solutions to Odd-Numbered Exercises. Index.

1,250 citations


"On the k-Mersenne–Lucas numbers" refers background in this paper

  • ...1 Introduction For a long time, many researchers have studied well-known number sequences like Fibonacci, Lucas, Pell, Jacobasthal, and Mersenne in order to get intrinsic theory and applications of these numbers in many research areas as physics, engineering, architecture, nature, and art [5]....

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01 Jan 2011
TL;DR: In this paper, the authors examined some of the interesting properties of the k-Lucas numbers themselves as well as looking at its close relationship with k-Fibonacci numbers.
Abstract: From a special sequence of squares of k-Fibonacci numbers, the kLucas sequences are obtained in a natural form. Then, we will study the properties of the k-Lucas numbers and will prove these properties will be related with the k-Fibonaci numbers. In this paper we examine some of the interesting properties of the k-Lucas numbers themselves as well as looking at its close relationship with the k-Fibonacci numbers. The k-Lucas numbers have lots of properties, similar to those of k-Fibonacci numbers and often occur in various formulae simultaneously with latter. Mathematics Subject Classification: 11B39, 11B83

90 citations

Journal ArticleDOI
TL;DR: The recurrence relations and the generating functions of the new family for k=2 and k=3 are described, and a few identity formulas for the family and the ordinary Fibonacci numbers are presented.

68 citations

Journal ArticleDOI
TL;DR: In this paper, the Binet's formula for k-Pell numbers was obtained and some properties of the binet formula were obtained for the k-pell numbers, as well as the generating function for the general term of the sequence, using the ordinary generating function.
Abstract: We obtain the Binet’s formula for k-Pell numbers and as a consequence we get some properties for k-Pell numbers. Also we give the generating function for k-Pell sequences and another expression for the general term of the sequence, using the ordinary generating function, is provided. Mathematics Subject Classification: 11B37, 05A15, 11B83.

55 citations

Journal ArticleDOI
21 Jan 2021
TL;DR: In this paper, Binet's formula, generating function and symmetric function of Mersenne Lucas numbers were given for products of (p, q)-numbers with Mersennes Lucas numbers at positive and negative indice.
Abstract: In this paper, we first introduce new definition of Mersenne Lucas numbers sequence as, for n > 2, mn = 3mn−1 − 2mn−2 with the initial conditions m0 = 2 and m1 = 3. Considering this sequence, we give Binet’s formula, generating function and symmetric function of Mersenne Lucas numbers. By using the Binet’s formula we obtain some well-known identities such as Catalan’s identity, Cassini’s identity and d’Ocagne’s identity. After that, we give some new generating functions for products of (p,q)-numbers with Mersenne Lucas numbers at positive and negative indice.

8 citations


"On the k-Mersenne–Lucas numbers" refers methods in this paper

  • ...[7] The Mersenne–Lucas numbers with negative index are given by m−n = 1 2n mn, for all n ∈ N....

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