Connected graphs by total distance

[franktaw at netscape.net]

For a connected graph, take the sum over all pairs of vertices of the 
minimal distance between them.  Let a(n) be the number of graphs for 
which this sum is n.  I think a(n) starts (from offset 1):

1,0,1,1,0,1,1,2,1,2

(a(0) could be either 1 or 2, depending on whether you want to count 
the empty graph.)

Looking only at trees, the sequence starts (from offset 0):

1,1,0,0,1,0,0,0,0,1,1,0,0,0,0,0,1,0,1,0,1,0,0,0,0,1

(I don't consider the empty graph to be a tree.)

If I haven't made any mistakes, neither of these sequences is in the 
OEIS.

Franklin T. Adams-Watters