Pascal's Triangle and Polygonal Numbers

Pascal's Triangle and Polygonal Numbers
By Alec L Shute

Polygonal numbers are a kind of general set of patters, a sequence of sequences. Common examples include triangle and square numbers, but we can also have less well known sequences such as pentagonal, hexagonal, heptagonal etc. numbers, all of which are closely linked with Pascal's triangle.

First I will explain how all of these sequences can be formed. Triangle numbers are made from adding consective integers, or adding one more each time. The first few terms are 1,3,6,10,15,21,28,36,45,55. To get to the next term, you add 2 then 3, then 4 and so on.

The square numbers are usually thought of as the sequence made from multiplying numbers by themselves, for example the sixth square is 6 x 6 = 36. However, for the purpose of linking them to triangle numbers and the other polygonal sequences, we shall consider them in a slightly different way. Square numbers can be made by adding consecutive odd numbers - the sequence 1,4,9,16,25,36,49... has differences of 3,5,7,9,11,13... , which are the odd numbers.

Continuing this idea, the pentagonal sequence is 1,5,12,22,35,51... which have a difference of 4,7,10,13,16... , which are the multiples of 3 add 1, and the hexagonal numbers are 1,6,15,28,45,66... , which have a difference of 5,9,13,17,21... , which are the multiples of 4 add 1 (the hexagonal sequence also turns out to be every other triangle number). So an n-gonal number will have a first term of 1, then differences corresponding to multiples of n-2 add 1.

Now we can link all this in with Pascal's triangle. The triangle numbers 1,3,6,10,15... are famously found in the third diagonal in of Pascal's triangle, as shown in bold below:

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 21 15 6 1

The square numbers (or any other polygonal sequences for that matter), however, are much harder to spot. The trick is to look in the same diagonal as we just obtained the triangle numbers from, but as they themselves don't appear there, we have to do a bit of adding to get them. The square numbers can be found by taking the sums of the consecutive values in this diagonal. So we get

(0) + 1 = 1

1 + 3 = 4

3 + 6 = 9

6 + 10 = 16 etc.

We apply a very similar process to create any polygonal sequence from Pascal's triangle. For the pentagonal numbers, we must multiply the first number by 2:

2 x (0) + 1 = 1

2 x 1 + 3 = 5

2 x 3 + 6 = 12

2 x 6 + 10 = 22 etc.

For hexagonal numbers, we multiply the first value in the sum by 3, for heptagonal numbers we multiply the first value by 4 and so on. This shows how we can create any polygonal number from Pascal's Triangle. This just goes to show how many patterns can be explored in Pascal's Triangle, as we have created an infinitely many sequences just from a single diagonal! For more information on some of the amazing patterns and properties of Pascal's Triangle, as well as a visual representation of the polygonal numbers, you are welcome to visit my site listed in the links below.

Pascal's Triangle

Pascal's Triangle

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Explaining the Links Between Pascal's Triangle and Sierpinski's Triangle

Explaining the Links Between Pascal's Triangle and Sierpinski's Triangle
By Alec L Shute

By assigning different colours to the odd and even numbers in Pascal's triangle, Sierpinski's triangle can be generated, as I have explained in my previous post entitled "Pascal's Triangle and Sierpinski's triangle". In this post, however, I attempt to explain why this interesting link arises between these two seemingly unrelated triangles, and how we can be sure that this pattern will always continue.

To begin with, as Pascal's triangle is a series of additions and we are colouring each number by whether it is odd or even, it seems sensible to look at some basic rules of adding odd and even numbers:

Rule 1: odd + odd = even,

Rule 2: 0dd + even or even + odd = odd

Rule 3: even + even = even

So, now to the main stage of the proof. Assume that a finite number of rows of Pascal's triangle did correspond to Sierpinski's triangle after a finite number of iterations. The bottom of Sierpinski's triangle is always a row of triangles all the same colour which correspond to odd numbers in Pascal's triangle. By rule number 1, the next row of Pascal's triangle will be all even (except outer 1s which are of course odd).

From the third rule, we can see that a row such as row eight, 1,8,28,56,70,56,28,1, which has many even numbers in a row will create an upside down triangle of even numbers below it - here, we have 6 even numbers in a row, and the 5 numbers in between them in the row below will be even, by rule 3, and the 4 in between those in the row below will be even and so on. So after our row of all odd numbers, we get a triangle of even numbers.

If you try this out, you should notice that the new triangle we just created is the same size as the section of Sierpinski's triangle we were just dealing with.

We now must look at what is going to happen due to those 1s on the edge of our otherwise even row. Will they not change what goes on in the rows below them? Yes, but the fact is that the very first number in Pascal's triangle is a 1, so why would the pattern below these 1s be any different to that first section of Sierpinski's triangle we had? Sure, the actual numbers will be different, but the parity (oddness/evenness) will be identical to before, as from rule 2, the parity of a number does not change by adding an even number to it.

Let's take a step back and look at what we have achieved so far. we now have a much bigger Sierpinski's triangle which is made up of 4 parts:

1. the initial section of Sierpinski's triangle that we started with

2. the all-even upside-down triangle below it.

3. The left hand side equilateral triangle that is identical in colouring to that in 1.

4. The right hand side triangle, again identical to 1.

Therefore, we have made the next iteration of the rule generating Sierpinski's triangle! You can check it works for the first iteration, then we could do this again and again and again and create whatever iteration of Sierpinski's we wanted. Do it infinitely, and we will thus create an infinite Pascal's triangle that yields the actual infinite fractal of Sierpinski's triangle itself. We are done! We have explained the link between Pascal's triangle and Sierpinski's triangle!

I know that these kinds of proofs can sometimes be rather hard to visualise. Please visit my site shown in the resource box below, which contains three articles each (with lots of pretty pictures and diagrams) all about Sierpinski's triangle and other fractals and their link with Pascal's triangle:

http://pascalangle.com

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