GoNumSquares.java

public class GoNumSquares {

    private static final int BOARD_SIZE = 19;

    public static void main(final String... args) {
        int squareSum = 0;
        for (int i = BOARD_SIZE; i > 0; i--) {
            final int dim = (BOARD_SIZE - i) + 1;
            squareSum += (dim * dim);
        }
        System.out.println(squareSum);
    }
}

Result: 2470