The progression displayed by the most used board sizes, 9x9, 13x13 and 19x19, suggests a generalization to what is believed to be those sizes of which "interesting" games can emerge:
7x7, 9x9 (+2), 13x13 (+4), 19x19 (+6), 27x27? (+8), 37x37? (+10), ...
This sequence can be captured by the formula n^2+n+7.
Bildstein: Let me be the first to disagree: Although you found a formula that correctly gives 9, 13 and 19, I think if anyone ever plays on a 27x27 or 37x37 board, they will find them not particularly interesting.
I think the reason for the 9,13,19 progression is this: 19x19 is an appropriate size, as described elsewhere on this site, 9x9 is approximately half the size of the 19x19, and 13x13 is approximately helf way between the two (as an intermediate step between playing on 9x9 and playing on a full sized board, perhaps).
Coconuts: Old games in Japan were played on a 17x17 board,[1] and they experimented with 21x21 boards as well. I think any board size can be interesting (even 2x2 if you're looking for the right thing), even or odd, small or large. Go is so adaptable that it almost doesn't matter (or you can look at some of SL's investigation into lineless go).
Rubyflame: Count the number of intersections. For 9x9, you have 81 intersections. For 13x13, you have 169 intersections (compare to 2*81 = 162). For 19x19, you have 361 intersections (compare to 2*169 = 338).
In each case, it's the closest odd-sized board you can get to twice the previous board size. This also suggests that a 13x13 game takes about twice as long as a 9x9 game, and a 19x19 game takes about twice as long as that.
By this formula, the next sizes are 27x27 and 39x39.
[1] Bill: I believe that a 17x17 go board was found in an archeological dig in China, dating from before go came to Japan.