Triangular Day: Huh?

Last year, we had a tradition where every Saturday whose date was a Fibonacci number, the post would be about Fibonacci numbers. Since my blog has been up for a year, I decided that we should change it up a little bit and look at a difference sequence of numbers.

I ended up choosing the triangular numbers: a sequence that is also simple to understand and has some really fascinating things about it. First off, what are they? Here are the first several.

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210

What is special about this sequence? Let's try to figure out a formula for this sequence where you plug in a number and it gives you the number in the sequence in that position. That is known as an explicit formula. We can find that by looking for common differences.

1     3     6     10     15     21
  \   /  \   /   \   /   \   /    \   /
   2      3      4      5       6
     \    /  \    /  \    /   \    /
       1      1      1       1

We found a common difference. You would then create a system to solve for a, b, and c: the variables in the quadratic equation (since it is second differences, the equation puts a number to the second power) that would be the formula. If you do that, you get the formula:

1/2n^2 + 1/2n

That looks a little ugly, but we can make it look nicer by going:

n(n + 1)/2

Okay, that's a cool formula. But that isn't so special, and this sequence obviously is or it wouldn't have it's own name.

Think about the name for a second; the triangular numbers. Let's think about triangles. If you were to make a triangular array (a triangle made up of dots) with one dot per side, it would look like this:

• = 1 dot

What about an array with 2 dots per side?

  •     = 3 dots
•   •

What about an array with 3 dots per side?

    •
  •   •    = 6 dots
•   •   •

How about 4 dots per side?

      •
    •   •
  •   •   •     = 10 dots
•   •   •   •

See the pattern? The number of dots in the triangular arrays make up the numbers of the sequence. Something you could use this formula for is to think if the back row of a bowling lane had 7 pins rather than 4, than how many pins is a strike/spare? Turns out, just plug it into the formula:

7(7 + 1)/2
7(8)/2
56/2
28

There would in fact be 28 pins in this lane.

One last pattern; look at the differences between all of the numbers in the sequence. You have:

2, 3, 4, 5, 6, 7, ...

For the seventh number, you would find it by adding seven to the sixth number. This formula is known as the recursive formula, which is written as:

T(n) = T(n-1) + n

There are so many more cool and easy to understand patterns involving this simple set of numbers that are just as surprising as the Fibonacci numbers. And lucky us, next week is a triangular day too!

June Problem of the Week answers:

Easy:

h = 2
a = 1
z = 4
n = 8
t = 1024
l = 16

Hard:

f = 53.1
g = 6
h = 10
a = 1
b = -21
c = 104
x1 = 8
x2 = 13
n = 6725
d = 82
z = 5.4

If you haven't already, make sure to do July's problem of the week.