Here is a math problem a friend asked for my help with: On a 2025x2025 chessboard, place some rooks on certain squares. Each rook can cover its row, its column, and its northwest-southeast diagonal. What is the minimum number of rooks required to ensure every square on the board is covered?

I wrote a DFS program and found a pattern for n=5, 7, 9, 11. It roughly looks like this (though this pattern might be wrong):
1 1
2 3 & 3 2
4 6 & 6 4
5 8 & 8 5
7 11 & 11 7
...

If we continue filling the board in this way, it should cover all squares of the 2025x2025 board. If I didn’t miscalculate, the number of rooks required under this method is 1547. Does anyone have a better solution?