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EXAMPLE
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1 X 1: 0 dots, since there are already no squares,
2 X 2: 1 dot, any vertex will do,
3 X 3: 2 dots, the center kills all the small squares and you need one corner to kill the big square,
a(4) = 4: there are 4 disjoint squares, so it must be at least 4, and with a little more work you can find a set of 4 dots that work.
From Robert Israel:
For the 5 X 5 case, Maple confirms that the optimal solution is 8 dots,
which can be placed at
[1, 1], [1, 3], [2, 2], [2, 3], [3, 0], [3, 1], [3, 2], [4, 4]
For the 6 X 6 case, Maple tells me that the optimal solution is 12 dots,
which can be placed at
[0, 5], [1, 1], [1, 2], [1, 4], [2, 0], [2, 3], [2, 4], [3, 1], [3, 3],
[4, 0], [4, 4], [5, 2]
For the 7 X 7 case, Maple tells me that the optimal solution is 17 dots,
which can be placed at
[0, 0], [1, 2], [1, 3], [1, 5], [2, 1], [2, 4], [2, 5], [3, 2], [3, 3],
[3, 4], [4, 1], [4, 2], [4, 5], [5, 1], [5, 3], [5, 4], [6, 6]
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