# Is there a name for this pattern involving the differences between perfect squares, cubes, etc?

I stumbled upon this pattern and was wondering if it has a name or any applications?

Take a set of consecutive perfect squares and find the difference between any consecutive pair. Then find the difference between each consecutive difference equals 2.

• 4 - 1 = 3
• 9 - 4 = 5 -> 5-3 = 2
• 16 - 9 = 7 -> 7-5 = 2
• 25 - 16 = 9 -> 9-7 = 2

Take a set of consecutive perfect cubes and find the difference between any consecutive pair. Then find the difference between the difference between each consecutive difference equals 6.

• 8 – 1 = 7
• 27 – 8 = 19 -> 19 - 7 = 12
• 64 – 27 = 37 -> 37 - 19 = 18 -> 18-12 = 6
• 125 – 64 = 61 -> 61 = 37 = 24 -> 24 -18 = 6

The same applies for x^4 but as you may suspect you must use another “layer of differences” before finding the common difference of 24.

You are hitting on an idea of calculus. What you are examining is the discrete derivatives of $x^n$. The n-th iteration will give you $n!$.