$3^2$ anti-magical squares doesn't exist!

A heterosquare contains positive consecutive integers starting from 1 such that the rows, columns, and diagonals all add to different values. If the sums resulting from a heterosquare form a consecutive sequence, the heterosquare is also called an anti-magic square.

Assume that a $3^2$ anti-magical square exists. It is already known that the minimum value of the sum of any row is 6, and the maximum value of the sum of any row is 24, so there are 12 sequences you have to consider, from $(6,7,...,13)$ to $(17,18,...,24)$ . Without using brute force (i.e. testing all heterosquares using a computer) or calculating the value of any row, column and diagonal, how many sequences could you eliminate?

If after eliminating, you still have $n$ sequences left, and the smallest value of every sequence is $a_1,a_2,...,a_n$ , find $n+a_1+a_2+...+a_n$ .

Note : If your answer is different, feel free to report, in any way.