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 (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 Triangle and Fibonacci Numbers

1991 ◽  
Vol 84 (4) ◽  
pp. 314-319
Author(s):  
James Varnadore
Keyword(s):  
Fibonacci Numbers ◽  
Pascal’S Triangle ◽  
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.

More power to Pascal

2008 ◽  
Vol 92 (525) ◽  
pp. 454-465 ◽  
Author(s):  
Barry Lewis
Keyword(s):  
Fibonacci Numbers ◽  
Ancient Greece ◽  
Binomial Coefficients ◽  
Medieval Italy ◽  
Pascal’S Triangle ◽  
Pascal's Triangle

Pascal’s triangle is the most famous of all number arrays - full of patterns and surprises. One surprise is the fact that lurking amongst these binomial coefficients are the triangular and pyramidal numbers of ancient Greece, the combinatorial numbers which arose in the Hindu studies of arrangements and selections, together with the Fibonacci numbers from medieval Italy. New identities continue to be discovered, so much so that their publication frequently excites no one but the discoverer.

Linear recurrences associated to rays in Pascal’s triangle and combinatorial identities

2014 ◽  
Vol 64 (2) ◽  
Author(s):  
Hacène Belbachir ◽  
Takao Komatsu ◽  
László Szalay
Keyword(s):  
Recurrence Relation ◽  
Generating Function ◽  
Finite Differences ◽  
Continued Fractions ◽  
Fibonacci Numbers ◽  
Combinatorial Identities ◽  
Bivariate Polynomials ◽  
Linear Recurrences ◽  
Pascal’S Triangle ◽  
Pascal's Triangle

AbstractOur main purpose is to describe the recurrence relation associated to the sum of diagonal elements laying along a finite ray crossing Pascal’s triangle. We precise the generating function of the sequence of described sums. We also answer a question of Horadam posed in his paper [Chebyshev and Pell connections, Fibonacci Quart. 43 (2005), 108–121]. Further, using Morgan-Voyce sequence, we establish the nice identity $F_{n + 1} - iF_n = i^n \sum\limits_{k = 0}^n {(_{2k}^{n + k} )( - 2 - i)^k } $ of Fibonacci numbers, where i is the imaginary unit. Finally, connections to continued fractions, bivariate polynomials and finite differences are given.

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 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.

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