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 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.

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 (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
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