So it seems that a 7x6 board was chosen because it's "the smallest board which isn't easily shown to be a draw". In addition, it was also speculated that there should probably be an even amount of columns. Therefore, it seems that the dimensions of a Connect $N$ board are a function of $N$. I see two possible functions:
N.B.: I'm not sure if there's a rule about the numbers being consecutive, but I'm assuming that that is the case here.
Times 1.5 function pseudo-code:
column_height = N * 1.5;
If column_height is an even number:
row_height = N + 1;
Otherwise (if column_height is an odd number):
column_height = (N * 1.5) + 1; //truncate the decimal portion of (N * 1.5) before adding one
row_height = column_height + 1;
Add 3 function psuedo-code:
column_height = N + 3
If column_height is an even number:
row_height = N + 2;
Otherwise (if column_height is an odd number):
column_height = N + 4;
row_height = N + 3;
The first one seems more likely, but since I'm trying to generate perfectly mathematically balanced game boards and there doesn't seem to be any symmetry that I can see, I'm still not sure. Does this seem about right?