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 7 8 9 10
19 6 1 2 11
18 5 4 3 12
17 16 15 14 13$1 + 3 + 5 + 7 + 9 + 13 + 17 + 21 + 25 = 101$
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?
When we know the corner of a certain spiral we can calculate it's total like $f_n = 4 c_n + 6 (n - 1)$. We then only have to update the corner value for each spiral.
total = 1
current_corner = 3
for n in range(3, 1002, 2):
total += 4 * current_corner + 6 * (n - 1)
current_corner += 4 * n - 2
s = total
assert(s == 669171001)
s
The only missing piece is how could we calculate the current corner value for a certain n. The series for this is as follows:
$c = 1, 3, 13, 31, 57, 91, 133, 183, 241, \dots$
With some experimenting it can be seen that
$c_n = (n - 1)^2 - (n - 2) = n^2 - 2n + 1 - n + 2 = n^2 - 3n + 3$.
Now, we can insert $c_n$ into $f_n$ which gives as the sum of corners for the nth spiral:
$f_n = 4n^2 - 12n + 12 + 6n - 6 = 4n^2 - 6n + 6$
s = 1 + sum([4 * n * n - 6 * n + 6 for n in range(3, 1002, 2)])
assert(s == 669171001)
s