Polygonal numbers are a kind of general set of patterns, a sequence of sequences. Common examples include triangular and square numbers, but we can also have lesser-known sequences like pentagonal, hexagonal, heptagonal, etc. numbers, all of which are closely related to Pascal’s triangle.

First I will explain how all these sequences can be formed. Triangular numbers are formed by adding consecutive integers or by adding one more at a 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.

Square numbers are generally thought of as the sequence obtained by multiplying numbers by themselves, for example, the sixth square is 6 x 6 = 36. However, in order to relate them to triangular numbers and other polygonal sequences, we will consider them in a slightly different way. Square numbers can be formed 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 with 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 plus 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 plus 1 (the hexagonal sequence also happens to be any other triangle number). So, an n-gonal number will have a first term of 1, so the differences corresponding to the multiples of n-2 add up to 1.

Now we can link all of this to Pascal’s triangle. The triangular numbers 1,3,6,10,15… lie on the third diagonal of Pascal’s triangle, as shown in bold below:

1

eleven

1 2 1

1 3 3 1

1 4 6 4 1

fifteen 10 10 5 1

sixteen fifteen 21 15 6 1

Square numbers (or any other polygonal sequence), however, are much more difficult to detect. The trick is to look on the same diagonal from which we just got the triangle numbers, but since they don’t appear there themselves, we have to do a bit of addition to get them. The square numbers can be found by taking the sums of the consecutive values ​​on 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 pentagonal numbers, we must multiply the first number by 2:

2x(0) + 1 = 1

2×1 + 3 = 5

2×3 + 6 = 12

2 x 6 + 10 = 22 etc.

For hexagonal numbers, we multiply the first value of 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, since we’ve created infinitely many sequences just from a single diagonal! To learn more about some of the amazing patterns and properties of Pascal’s Triangle, as well as a visual representation of polygonal numbers, you can visit my site found at the links below.