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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