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Pascal’s Triangle and Polygonal Numbers
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 such as pentagonal, hexagonal, heptagonal numbers, etc., all of which are closely related to Pascal’s triangle.
I will first explain how all these sequences can be formed. Triangular numbers are created by adding consecutive whole numbers or adding one more each time. The first terms are 1,3,6,10,15,21,28,36,45,55. To move on to the next term, you add 2 then 3, then 4 and so on.
Square numbers are generally considered to be the sequence consisting of numbers that are multiplicative by themselves, for example the sixth square is 6 x 6 = 36. However, for the purpose of relating them to triangular numbers and other polygonal sequences, we will consider them in a slightly different way. Square numbers can be created 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 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 turns out to be every other triangle number). Thus, an n-gonal number will have a first term of 1, then the differences corresponding to the multiples of n-2 add 1.
Now we can relate all of this with Pascal’s triangle. The triangle numbers 1,3,6,10,15… lie in the third diagonal of Pascal’s triangle, as shown in bold below:
1 2 1
1 3 3 1
1 4 6 4 1
1 5 ten 10 5 1
1 6 15 21 15 6 1
Square numbers (or any other polygonal sequence for that matter), however, are much harder to spot. The trick is to look in the same diagonal from which we just got the triangle numbers, but since they aren’t there themselves, we have to do a bit of addition to get them. Square numbers can be found by taking the sums of 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 pentagonal numbers, 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 shows how many patterns can be explored in Pascal’s triangle, because we have created an infinite number of sequences 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 polygonal numbers, you are welcome to visit my site listed in the links below.
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