Pascal's Triangle— a Serendipitous Source for Programming Activities

1983 ◽  
Vol 76 (9) ◽  
pp. 686-690
Author(s):  
Marita Eng ◽  
John Casey
Keyword(s):  
Number Theory ◽  
Computer Programming ◽  
Pascal’S Triangle ◽  
Pascal's Triangle

Pascal's triangle is an elegant array of numbers that has fascinated mathematicians since its invention. Its applications in algebra, number theory, and probability are well known. The use of Pascal's triangle in the classroom need not be restricted to these conventionally accepted applications. Explorations of Pascal's triangle through the use of computer programming create situations that are incredibly rich in their mathematical yield.

Programming Pascal's Triangle

1974 ◽  
Vol 67 (8) ◽  
pp. 705-708
Author(s):  
Walter Curley
Keyword(s):  
Computer Programming ◽  
Pascal’S Triangle ◽  
Pascal's Triangle

Computer programming can be used to extend a textbook topic.

scholarly journals On a Generalization of a Lucas’ Result and an Application to the 4-Pascal’s Triangle

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 288
Author(s):  
Atsushi Yamagami ◽  
Kazuki Taniguchi
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Prime Number ◽  
Special Issue ◽  
Pascal’S Triangle ◽  
Mersenne Number ◽  
Modulo P ◽  
Pascal's Triangle ◽  
Positive Integers ◽  
Symmetric Property

The Pascal’s triangle is generalized to “the k-Pascal’s triangle” with any integer k ≥ 2 . Let p be any prime number. In this article, we prove that for any positive integers n and e, the n-th row in the p e -Pascal’s triangle consists of integers which are congruent to 1 modulo p if and only if n is of the form p e m − 1 p e − 1 with some integer m ≥ 1 . This is a generalization of a Lucas’ result asserting that the n-th row in the (2-)Pascal’s triangle consists of odd integers if and only if n is a Mersenne number. As an application, we then see that there exists no row in the 4-Pascal’s triangle consisting of integers which are congruent to 1 modulo 4 except the first row. In this application, we use the congruence ( x + 1 ) p e ≡ ( x p + 1 ) p e − 1 ( mod p e ) of binomial expansions which we could prove for any prime number p and any positive integer e. We think that this article is fit for the Special Issue “Number Theory and Symmetry,” since we prove a symmetric property on the 4-Pascal’s triangle by means of a number-theoretical property of binomial expansions.

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

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''.

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