In this paper, we study a generalization of Narayana’s numbers and Padovan’s numbers. This generalization also includes a sequence whose elements are Fibonacci numbers repeated three times. We give combinatorial interpretations and a graph interpretation of these numbers. In addition, we examine matrix generators and determine connections with Pascal’s triangle.

/pdf/on-f-3-k-n-f-3-k-n-numbers-of-the-fibonacci-type-2husgqowf2.pdf

On $$F_3(k,n)$$ F 3 ( k , n ) -numbers of the Fibonacci type




UDC 517.5



We define 

 d

-Gaussian Fibonacci polynomials and 

 d

-Gaussian Lucas polynomials. We present the matrix representations of these polynomials. By using the Riordan method, we obtain the factorizations of the Pascal matrix including the polynomials. In addition, we define the infinite 

 d

-Gaussian Fibonacci polynomial matrix and the 

 d

-Gaussian Lucas polynomial matrix and give their inverses.



 d

-Gaussian Fibonacci, 

 d

-Gaussian Lucas polynomials and their matrix representations

We give the bicomplex Gaussian Fibonacci and the bicomplex Gaussian Lucas numbers and establish the generating functions and Binet’s formulas related to these numbers. Also, we present the summation formula, matrix representation and Honsberger identity and their relationship between these numbers. Finally, we show the relationships among the bicomplex Gaussian Fibonacci, the bicomplex Gaussian Lucas, Gaussian Fibonacci, Gaussian Lucas and Fibonacci numbers.

/pdf/on-the-bicomplex-gaussian-fibonacci-and-gaussian-lucas-2owxny8j.pdf

On the bicomplex Gaussian Fibonacci and Gaussian Lucas numbers

Bu çalışmanın amacı Gauss Bronz Lucas sayı dizisini tanıtmak ve incelemektir. İlk olarak Bronz Lucas sayılarını genişleterek Gauss Bronz Lucas sayılarını tanımladık. Daha sonra bu sayı dizisi için Binet formülü ve üreteç fonksiyonunu bulduk. Ayrıca Gauss Bronz Lucas sayıları ile ilgili bazı toplam formülleri ve matrisleri araştırdık. Son olarak, bu dizinin Binet formülünü dikkate alarak Catalan, Cassini ve d’Ocagne özdeşlikleri gibi bilinen eşitlikleri elde ettik.

/pdf/gauss-bronz-lucas-sayilari-3scr2yp7.pdf

Gauss Bronz Lucas Sayıları

In this paper, inspiring Hosoya’s triangle, we define a new Narayana triangle. Then, we represent this Narayana triangle geometrically on the plane. In addition, we give some identities and properties of the new Narayana triangle.

New narayana triangle

We give a new definition of Narayana polynomials and show that there is a relationship between the coefficient of the new Narayana polynomials and Pascal’s triangle. We define the Gauss Narayana nu...

On the new Narayana polynomials, the Gauss Narayana numbers and their polynomials

In this paper, an expansion of the classical hyperbolic functions is presented and studied. Also, many features of the [Formula: see text]-Jacobsthal hyperbolic functions are given. Finally, we introduced some graph and curved surfaces related to the [Formula: see text]-Jacobsthal hyperbolic functions.

Hyperbolic functions obtained from k-Jacobsthal sequences

/pdf/the-k-fibonacci-group-and-periods-of-the-k-step-fibonacci-gzhhn1a1.pdf

The k-Fibonacci Group and Periods of the k-step Fibonacci Sequences

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.

/pdf/on-the-mersenne-and-mersenne-lucas-hybrinomial-quaternions-2tzzhhu6.pdf

On the Mersenne and Mersenne-Lucas hybrinomial quaternions

Abstract In this paper, we bring into light, study the polygonal structure of Fibonacci polynomials that are placed clockwise on these by a number corresponding to each vertex. Also, we find the relation between the numbers with such vertices. We present a relation for obtained sequence in an n-gon yielding the m-th term formed at k vertices. Also, we apply these situations to Lucas polynomials and find new recurrence relations. Then, the numbers obtained by writing the coefficients of these polynomials in step form are shown in OEIS.

Bahar Kuloğlu

Papers

On the new Narayana polynomials, the Gauss Narayana numbers and their polynomials

Hyperbolic functions obtained from k-Jacobsthal sequences

The k-Fibonacci Group and Periods of the k-step Fibonacci Sequences

On the Mersenne and Mersenne-Lucas hybrinomial quaternions

Fibonacci and Lucas Polynomials in n-gon