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 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.

scholarly journals Distance Fibonacci Polynomials—Part II

Symmetry ◽  
2021 ◽  
Vol 13 (9) ◽  
pp. 1723
Author(s):  
Urszula Bednarz ◽  
Małgorzata Wołowiec-Musiał
Keyword(s):  
Generating Function ◽  
Initial Conditions ◽  
Symmetric Matrix ◽  
Fibonacci Polynomials ◽  
Lucas Polynomials ◽  
Graph Interpretation ◽  
Q Matrix ◽  
Generalized Lucas Polynomials ◽  
Symmetric Patterns

In this paper we use a graph interpretation of distance Fibonacci polynomials to get a new generalization of Lucas polynomials in the distance sense. We give a direct formula, a generating function and we prove some identities for generalized Lucas polynomials. We present Pascal-like triangles with left-justified rows filled with coefficients of these polynomials, in which one can observe some symmetric patterns. Using a general Q-matrix and a symmetric matrix of initial conditions we also define matrix generators for generalized Lucas polynomials.

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

2020 ◽  
pp. 2150100
Author(s):  
Engi̇n Özkan ◽  
Bahar Kuloğlu
Keyword(s):  
Hankel Transform ◽  
Pell Numbers ◽  
Pascal’S Triangle ◽  
Narayana Numbers ◽  
Definition Of ◽  
Pascal's Triangle ◽  
The Relationship ◽  
Derivatives Of

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 numbers and their polynomials. Then we show that there is a relationship between the Gauss Narayana polynomials and the new Narayana polynomials. Also, we show that there is a relationship between the derivatives of the new Narayana polynomials and Pascal’s triangle. We also explain the relationship between the new Narayana polynomials and the known Pell numbers. Finally, we give the Hankel transform of the new Narayana polynomials.

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.

RECURRENT TWO-DIMENSIONAL SEQUENCES GENERATED BY HOMOMORPHISMS OF FINITE ABELIAN p-GROUPS WITH PERIODIC INITIAL CONDITIONS

Fractals ◽  
2011 ◽  
Vol 19 (04) ◽  
pp. 431-442 ◽  
Author(s):  
MIHAI PRUNESCU
Keyword(s):  
Initial Conditions ◽  
Two Dimensional ◽  
Pascal’S Triangle ◽  
Modulo P ◽  
Pascal's Triangle ◽  
Context Free

We prove that if a recurrent two-dimensional sequence with periodic initial conditions coincides in a sufficiently large starting square with a two-dimensional sequence produced by an expansive system of context-free substitutions, then they must coincide everywhere. We apply this result for some examples built up by homomorphisms of finite abelian p-groups, in particular for Pascal's Triangle modulo pk, Pascal's Triangles modulo 2 with non-trivial periodic borders, and Sierpinski's Carpets with non-trivial periodic border. All these particular cases justify the conjecture that recurrent two-dimensional sequences generated by homomorphisms of finite abelian p-groups with periodic initial conditions can always be alternatively generated by expansive systems of context-free substitutions.

scholarly journals Computing Sticks against Random Walk

2019 ◽  
Vol 2 (1) ◽  
Keyword(s):  
Random Walk ◽  
Deterministic Model ◽  
Initial Conditions ◽  
Probabilistic Approach ◽  
Geometric Constructions ◽  
Pascal’S Triangle ◽  
Pascal Triangle ◽  
Usual Form ◽  
Recursive Formulas ◽  
Pascal's Triangle

A new deterministic model with the help of geometric constructions and computing sticks (not related to trajectories) is proposed for the new justification of consistency of the probabilistic approach to explain the random walk on a plane. A new, stepped form of the arithmetic triangle of Pascal based on the construction of horizontal and vertical lines (arrows) is suggested, a comparison is made with Pascal’s triangle of the usual form. A two-sided generalization of Pascal’s triangle is proposed. Geometric constructions and formulas for calculating the coefficients that fill in these new geometric (arithmetic) figures are given. Further types of generalization of the step-shaped Pascal triangle are proposed. Examples of generalized initial conditions and generalized recursive formulas for constructing various types of a generalized Pascal triangle are given.

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.

Sign in / Sign up

Export Citation Format

Share Document