Triangular Day: The Oblong Numbers

I don't know if you noticed, but today is a triangular day. It is April 6th, and six is the third triangular number.

You might remember from December when I brought up figurative families. The triangular numbers are a figurative family because each number can be represented by a triangular array of dots. Each array forms a geometric figure.

I also talked about square numbers as being a figurative family. Each number in that sequence can be represented by a square array, which is another geometric figure.

What about a rectangle? Considering that any number can be turned into a rectangular array, it would not be as interesting of a family. However, what about a rectangle whose array has just one more row than it does columns. For instance:

•  •  •  •
•  •  •  •
•  •  •  •

This has four rows and three columns, making it fit into this family. These numbers are called the "oblong numbers."

The oblong numbers go as follows:

2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132...

Here is a little property of oblong numbers. Divide each number in that sequence by two. You get:

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66...

You end up with the triangular numbers. Why is this? Remember the formula used to find any triangular number?

T_n = n(n + 1)/2

We said that a number is oblong if its array had one more row than it had columns. So, an oblong number must be factorable into two numbers that are just one apart. We can denote these two factors as n and n+1. This makes the oblong number equal to:

O_n = n(n + 1)

After dividing by two, we ended up with the triangular numbers.

I enjoy trying to find ways to combine these different families together. For instance, take the first triangular number, first square number, and first oblong number. This would be 1, 1, and 2 respectively. We will add the first two together, and then subtract the last one.

1 + 1 - 2 = 0

Let's try that with some more:

T₁ + S₁ - O₁ =     1 + 1 - 2         = 0
T₂ + S₂ - O₂ =     3 + 4 - 6         = 1
T₃ + S₃ - O₃ =     6 + 9 - 12       = 3
T₄ + S₄ - O₄ =     10 + 16 - 20   = 6
T₅ + S₅ - O₅ =     15 + 25 - 30   = 10
T₆ + S₆ - O₆ =     21 + 36 - 42   = 15

Do you see the pattern? Each of the answers is a triangular number. In fact, it is one less than the triangular number used in the example.

Why is this? When I saw it, I first thought that we could check by using algebra. Take the explicit formula for each sequence:

T_n = n(n + 1)/2
_Sn = n²
_On = n(n + 1)

Our goal is to end with T_n-1, so we will use n(n - 1)/2 to denote that.

[n(n + 1)/2] + [n²] - [n(n + 1)] = [n(n - 1)/2]

Let's simplify all of the brackets so we don't have parentheses to work with.

[n(n + 1)/2] + [n²] - [n(n + 1)] = [n(n - 1)/2]
[1/2n² + 1/2n] + [n²] - [n² + n] = [1/2n² - 1/2n]

Now, we will combine like terms.

[1/2n² + n² - n²] + [1/2n - n] = 1/2n² - 1/2n
1/2n² - 1/2n = 1/2n² - 1/2n

And there is our proof. Though oblong numbers are not the family of a regular polygon, it is still an interesting sequence to look at.