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 7 8 9 10
19 6 1 2 11
18 5 4 3 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?

从 $1$ 开始，按顺时针顺序铺开的 $5 \times 5$ 螺旋数阵如下所示：

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

可以看出，该数阵对角线上的数之和是 $101$ 。

考虑以同样方式构成的 $1001 \times 1001$ 螺旋数阵，其对角线上的数之和是多少？