Triangular Day... times eight plus one

I don't know if you noticed, but today is a triangular day! It is the third of November and three is a triangular number. Next week is also a triangular day, and I will be mentioning probably my favorite thing about them.

Let me quickly list some triangular numbers.

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120,...

They are basically the sum of consecutive natural numbers. So, 1 = 1, 3 = 1+2, 6 = 1+2+3, and so on.

This week, I wanted to try something. In August, we tried doing a triangular number times nine plus one. That resulted in other triangular numbers, which was fairly cool.

Today, let's try taking that down a notch. We will multiply by eight and add one.

1 x 8 + 1 = 9
3 x 8 + 1 = 25
6 x 8 + 1 = 49
10 x 8 + 1 = 81
15 x 8 + 1 = 121
21 x 8 + 1 = 169
28 x 8 + 1 = 225

Do you see any pattern? This is a little bit harder than the rest, but look closely.

They are all square numbers! It is fascinating to me that triangular numbers generate squares on their own, but it is true.


1 x 8 + 1 = 9 = 3^2
3 x 8 + 1 = 25 = 5^2
6 x 8 + 1 = 49 = 7^2
10 x 8 + 1 = 81 = 9^2
15 x 8 + 1 = 121 = 11^2
21 x 8 + 1 = 169 = 13^2
28 x 8 + 1 = 225 = 15^2

How do we know this goes on forever? Well, let's try to prove it. We know that the triangular numbers follow the formula n(n+1)/2. You can click here to see the post where that was proved.

Also, notice the numbers being squared. They are 3, 5, 7, 9, 11, and so on, which are all odd numbers. They fit into the formula:

2n + 1.

So, we can set up the following equation, considering n is what term we are using in the sequence. So, for the second triangular number, n = 2.

8(n(n + 1)/2) + 1 = (2n + 1)^2

Now, we will multiply through on both sides, so there are no parentheses.

8(n(n + 1)/2) + 1 = (2n + 1)(2n + 1)
8(n^2 + n)/2 + 1 = 4n^2 + 2n + 2n + 1
4n^2 + 4n + 1 = 4n^2 + 4n + 1

And already, we see that the two things are equal.

I found this pattern especially cool because of the switch between sequences, which Fibonacci numbers do all of the time. Probably next spring, I will be talking about the fact that triangulars and squares are much closer related than you'd think. Just look at their names and you should be able to see what I mean.