scholarly journals On Some Multinomial Sums Related to the Fibonacci Type Numbers

2020 ◽  
Vol 77 (1) ◽  
pp. 99-108
Author(s):  
Iwona Włoch ◽  
Andrzej Włoch
Keyword(s):  
Linear Recurrence ◽  
Companion Matrix ◽  
Order Linear ◽  
Pascal’S Triangle ◽  
Graph Interpretation ◽  
Pascal's Triangle

AbstractIn this paper we investigate Fibonacci type sequences defined by kth order linear recurrence. Based on their companion matrix and its graph interpretation we determine multinomial and binomial formulas for these sequences. Moreover we present a graphical rule for calculating the words of these sequences from the Pascal’s triangle.

scholarly journals Distance Fibonacci Polynomials by Graph Methods

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2075
Author(s):  
Dominik Strzałka ◽  
Sławomir Wolski ◽  
Andrzej Włoch
Keyword(s):  
Initial Conditions ◽  
Symmetric Matrix ◽  
Fibonacci Polynomials ◽  
Pascal’S Triangle ◽  
Graph Interpretation ◽  
Symmetric Structure ◽  
Binomial Formula ◽  
Narayana Numbers ◽  
Pascal's Triangle

In this paper we introduce and study a new generalization of Fibonacci polynomials which generalize Fibonacci, Jacobsthal and Narayana numbers, simultaneously. We give a graph interpretation of these polynomials and we obtain a binomial formula for them. Moreover by modification of Pascal’s triangle, which has a symmetric structure, we obtain matrices generated by coefficients of generalized Fibonacci polynomials. As a consequence, the direct formula for generalized Fibonacci polynomials was given. In addition, we determine matrix generators for generalized Fibonacci polynomials, using the symmetric matrix of initial conditions.

scholarly journals On $$F_3(k,n)$$-numbers of the Fibonacci type

2021 ◽  
Vol 27 (3) ◽  
Author(s):  
Engin Özkan ◽  
Nur Şeyma Yilmaz ◽  
Andrzej Włoch
Keyword(s):  
Fibonacci Numbers ◽  
Pascal’S Triangle ◽  
Graph Interpretation ◽  
Combinatorial Interpretations ◽  
Pascal's Triangle

AbstractIn 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.

scholarly journals Distance Fibonacci Polynomials

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1540
Author(s):  
Urszula Bednarz ◽  
Małgorzata Wołowiec-Musiał
Keyword(s):  
Generating Function ◽  
Fibonacci Polynomials ◽  
Pascal’S Triangle ◽  
Graph Interpretation ◽  
Pascal's Triangle

In this paper, we introduce a new kind of generalized Fibonacci polynomials in the distance sense. We give a direct formula, a generating function and matrix generators for these polynomials. Moreover, we present a graph interpretation of these polynomials, their connections with Pascal’s triangle and we prove some identities for them.

scholarly journals (2, k)-Distance Fibonacci Polynomials

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 303
Author(s):  
Dorota Bród ◽  
Andrzej Włoch
Keyword(s):  
Fibonacci Numbers ◽  
Fibonacci Polynomials ◽  
Pascal’S Triangle ◽  
Graph Interpretation ◽  
Natural Extensions ◽  
Pascal's Triangle

In this paper we introduce and study (2,k)-distance Fibonacci polynomials which are natural extensions of (2,k)-Fibonacci numbers. We give some properties of these polynomials—among others, a graph interpretation and matrix generators. Moreover, we present some connections of (2,k)-distance Fibonacci polynomials with Pascal’s triangle.

Pascal's Prism

2012 ◽  
Vol 96 (536) ◽  
pp. 213-220
Author(s):  
Harlan J. Brothers
Keyword(s):  
Probability Theory ◽  
Fractal Geometry ◽  
Euclidean Geometry ◽  
Fibonacci Series ◽  
Pascal’S Triangle ◽  
Number Sequences ◽  
Definition Of ◽  
Image Position ◽  
Pascal's Triangle

Pascal's triangle is well known for its numerous connections to probability theory [1], combinatorics, Euclidean geometry, fractal geometry, and many number sequences including the Fibonacci series [2,3,4]. It also has a deep connection to the base of natural logarithms, e [5]. This link to e can be used as a springboard for generating a family of related triangles that together create a rich combinatoric object.2. From Pascal to LeibnizIn Brothers [5], the author shows that the growth of Pascal's triangle is related to the limit definition of e.Specifically, we define the sequence sn; as follows [6]:

3057. A Note on Pascal's Triangle

1963 ◽  
Vol 47 (359) ◽  
pp. 57
Author(s):  
Robert Croasdale
Keyword(s):  
Pascal’S Triangle ◽  
Pascal's Triangle

scholarly journals The Fibonacci p-numbers and Pascal’s triangle

2016 ◽  
Vol 3 (1) ◽  
pp. 1264176 ◽  
Author(s):  
Kantaphon Kuhapatanakul ◽  
Lishan Liu
Keyword(s):  
Pascal’S Triangle ◽  
Pascal's Triangle
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