Pascal’s Triangle and Divisibility
نویسنده
چکیده
In this chapter we will be investigating the intricacies of a seemingly innocuous mathematical object Pascal’s Triangle. After a brief refresher of our understanding of the triangle, we will delve into Cellular Automata as a method of building patterns, and then relate this to a specific pattern within Pascal’s Triangle. Then we shall investigate specific divisibility identities that can aid in our discovery on patterns within the triangle. Finally, we will answer a long-asked question: what proportion of Pascal’s Triangle is made of odd numbers.
منابع مشابه
The generalization of Pascal’s triangle from algebraic point of view
In this paper we generalize Pascal’s Triangle and examine the connections between the generalized triangles and powering integers and polynomials respectively. The interesting and really romantic Pascal’s Triangle is a favourite research field of mathematicians for a very long time. The table of binomial coefficients has been named after Blaise Pascal, a French scientist, but was known already ...
متن کاملGeneralization of an Identity of Andrews
Abstract We consider an identity relating Fibonacci numbers to Pascal’s triangle discovered by G. E. Andrews. Several authors provided proofs of this identity, all of them rather involved or else relying on sophisticated number theoretical arguments. We not only give a simple and elementary proof, but also show the identity generalizes to arrays other than Pascal’s triangle. As an application w...
متن کاملStar Theorem Patterns Relating to 2n-gons in Pascal’s Triangle — and More
We first pose a Sudoku-type puzzle, involving lattice points in Pascal’s Triangle, and lines passing through them. Then, through the use of an expanded notation for the binomial coefficient (
متن کاملPascal’s triangle and other number triangles in Clifford Analysis
The recent introduction of generalized Appell sequences in the framework of Clifford Analysis solved an open question about a suitable construction of power-like monogenic polynomials as generalizations of the integer powers of a complex variable. The deep connection between Appell sequences and Pascal’s triangle called also attention to other number triangles and, at the same time, to the cons...
متن کاملPascal’s Triangles in Abelian and Hyperbolic Groups
We are used to imagining Pascal’s triangle as extending forever downwards from a vertex located at the top. But it is interesting to see it as occupying the first quadrant of the plane with it’s vertex at (0, 0). Imagine further that the plane is made of graph paper — that is, that we have embedded into it the Cayley graph of Z × Z with respect to the standard generating set. If we place the en...
متن کامل