scholarly journals Arithmetic Triangle

2017 ◽  
Vol 9 (2) ◽  
pp. 100
Author(s):  
Luis Dias Ferreira
Keyword(s):  
Arithmetic Progression ◽  
Modus Operandi ◽  
Interesting Point ◽  
Pascal’S Triangle ◽  
Initial Term ◽  
The Common ◽  
Pascal's Triangle

The product of the first $n$ terms of an arithmetic progression may be developed in a polynomial of $n$ terms. Each one of them presents a coefficient $C_{nk}$ that is independent from the initial term and the common difference of the progression. The most interesting point is that one may construct an "Arithmetic Triangle'', displaying these coefficients, in a similar way one does with Pascal's Triangle. Moreover, some remarkable properties, mainly concerning factorials, characterize the Triangle. Other related `triangles' -- eventually treated as matrices -- also display curious facts, in their linear \emph{modus operandi}, such as successive "descendances''.

scholarly journals Tree-Antimagicness of Disconnected Graphs

2015 ◽  
Vol 2015 ◽  
pp. 1-4 ◽  
Author(s):  
Martin Bača ◽  
Zuzana Kimáková ◽  
Andrea Semaničová-Feňovčíková ◽  
Muhammad Awais Umar
Keyword(s):  
Arithmetic Progression ◽  
Disjoint Union ◽  
Simple Graph ◽  
Initial Term ◽  
Vertex Set ◽  
The Common ◽  
Disconnected Graphs

A simple graphGadmits anH-covering if every edge inE(G)belongs to a subgraph ofGisomorphic toH. The graphGis said to be (a,d)-H-antimagic if there exists a bijection from the vertex setV(G)and the edge setE(G)onto the set of integers1, 2, …,VG+E(G)such that, for all subgraphsH′ofGisomorphic toH, the sum of labels of all vertices and edges belonging toH′constitute the arithmetic progression with the initial termaand the common differenced.Gis said to be a super (a,d)-H-antimagic if the smallest possible labels appear on the vertices. In this paper, we study super tree-antimagic total labelings of disjoint union of graphs.

Discovering Patterns in Pascal's Triangle

2019 ◽  
Vol 24 (4) ◽  
pp. 247-254
Author(s):  
Marilyn Howard
Keyword(s):  
Middle School ◽  
Problem Solving ◽  
School Level ◽  
Blaise Pascal ◽  
Pascal’S Triangle ◽  
The Common ◽  
Pascal's Triangle ◽  
Common Era ◽  
Middle School Level

The less you know about the patterns in Pascal's triangle, the more fun you will have discovering the triangle's many secrets. I am amazed at how few students and even teachers (especially at the middle school level) have ever explored Pascal's triangle. Although this famous triangle bears the name of Blaise Pascal (1623-1662), who saw many of the patterns when he was only thirteen years old, it had been around for centuries before he was born. See the ancient diagram in figure 1, which appeared at the front of a Chinese book in 1303 (Vakil 2008). Evidence suggests that the properties of the elements of Pascal's triangle were known before the common era. Students and teachers alike can enjoy exploring patterns through problem solving with Pascal's triangle.

On the Diophantine Equation n(n + d) · · · (n + (k − 1)d) = byl

2004 ◽  
Vol 47 (3) ◽  
pp. 373-388 ◽  
Author(s):  
K. Győry ◽  
L. Hajdu ◽  
N. Saradha
Keyword(s):  
Arithmetic Progression ◽  
Diophantine Equation ◽  
Arithmetic Progressions ◽  
Rational Solutions ◽  
Initial Term ◽  
The Common ◽  
Integral Solutions

AbstractWe show that the product of four or five consecutive positive terms in arithmetic progression can never be a perfect power whenever the initial term is coprime to the common difference of the arithmetic progression. This is a generalization of the results of Euler and Obláth for the case of squares, and an extension of a theorem of Győry on three terms in arithmetic progressions. Several other results concerning the integral solutions of the equation of the title are also obtained. We extend results of Sander on the rational solutions of the equation in n, y when b = d = 1. We show that there are only finitely many solutions in n, d, b, y when k ≥ 3, l ≥ 2 are fixed and k + l > 6.

ON -ANTIMAGICNESS OF DISCONNECTED GRAPHS

2016 ◽  
Vol 94 (2) ◽  
pp. 201-207 ◽  
Author(s):  
MARTIN BAČA ◽  
MIRKA MILLER ◽  
JOE RYAN ◽  
ANDREA SEMANIČOVÁ-FEŇOVČÍKOVÁ
Keyword(s):  
Arithmetic Progression ◽  
Disjoint Union ◽  
Simple Graph ◽  
Initial Term ◽  
The Common ◽  
Disconnected Graphs

A simple graph $G=(V,E)$ admits an $H$-covering if every edge in $E$ belongs to at least one subgraph of $G$ isomorphic to a given graph $H$. Then the graph $G$ is $(a,d)$-$H$-antimagic if there exists a bijection $f:V\cup E\rightarrow \{1,2,\ldots ,|V|+|E|\}$ such that, for all subgraphs $H^{\prime }$ of $G$ isomorphic to $H$, the $H^{\prime }$-weights, $wt_{f}(H^{\prime })=\sum _{v\in V(H^{\prime })}f(v)+\sum _{e\in E(H^{\prime })}f(e)$, form an arithmetic progression with the initial term $a$ and the common difference $d$. When $f(V)=\{1,2,\ldots ,|V|\}$, then $G$ is said to be super $(a,d)$-$H$-antimagic. In this paper, we study super $(a,d)$-$H$-antimagic labellings of a disjoint union of graphs for $d=|E(H)|-|V(H)|$.

1519. An extension of Pascal’s triangle.

1941 ◽  
Vol 25 (264) ◽  
pp. 118
Author(s):  
G. A. Garreau
Keyword(s):  
Pascal’S Triangle ◽  
Pascal's Triangle

Pascal's Triangle and the Tower of Hanoi

1992 ◽  
Vol 99 (6) ◽  
pp. 538-544 ◽  
Author(s):  
Andreas M. Hinz
Keyword(s):  
Tower Of Hanoi ◽  
Pascal’S Triangle ◽  
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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