Project Euler, Problem 28: Number Spiral Diagonals

Starting with the number 1 and moving to the right in a clockwise direction a 5 by 5 spiral is formed as follows:

21 22 23 24 25

20 07 08 09 10

19 06 01 02 11

18 05 04 03 12

17 16 15 14 13

It can be verified that the sum of the numbers on the diagonals is 101.

What is the sum of the numbers on the diagonals in a 1001 by 1001 spiral formed in the same way?

Right away I saw that there was a regularity to the numbers on the corners, especially the one on the top right. It was always equal to $n^{2}$ where $n$ was the length of the sides. The bottom left corner was equal to ${(n - 1)}^{2} + 1$ or $n^{2} - 2 n + 2$ . The top left corner was $n^{2} - n - 1$ and the bottom right corner was equal to $n^{2} - 3 n + 3$ .

Once I had all of that figured out, I ran a for loop counting up by 2s from 3 to 1,001, summing the corners and adding it to a running sum.

106,320