Cross flips (noch nicht übersetzt)

N×N disks are placed on a square game board. Each disk has a black side and white side.

At each turn, you may choose a disk and flip all the disks in the same row and the same column as this disk: thus 2×N-1 disks are flipped. The game ends when all disks show their white side. The following example shows a game on a 5×5 board.

It can be proven that 3 is the minimal number of turns to finish this game.

The bottom left disk on the N×N board has coordinates (0,0);
the bottom right disk has coordinates (N-1,0) and the top left disk has coordinates (0,N-1).

Let C_N be the following configuration of a board with N×N disks:
A disk at (x,y) satisfying $N - 1 \le \sqrt{x^2 + y^2} \lt N$, shows its black side; otherwise, it shows its white side. C₅ is shown above.

Let T(N) be the minimal number of turns to finish a game starting from configuration C_N or 0 if configuration C_N is unsolvable.
We have shown that T(5)=3. You are also given that T(10)=29 and T(1 000)=395253.

Find $\sum \limits_{i = 3}^{31} {T(2^i - i)}$.