Can someone save my sanity: I can't believe this isn't in the OEIS, I must have made a mistake.
"a(n) = the first column of Pascal's triangle in which n does not appear".
Labelling the first column 0 and starting at n=2, I get
2,3,2,2,3,2,2,2,4,2,2,2,2,3,2,2,2,2,2,3,2,2,2,2,...
In reply to @christianp
The columns of Pascal's triangle are
1,1,1,1,...
1,2,3,4,...
1,3,6,10,...
1,4,10,20,...
...
In reply to @christianp
So a(5) = 2 because it doesn't appear in column 2, which is 1,3,6,10,...