a(n) = number of strings (including the empty string) over an alphabet of size n that do not have any substrings of length > 1 that appear more than once in the string.

Here "substrings" have no "gaps", i.e. a substring means a subsequence of characters from the original string using contiguous indices.

The number of De Bruijn sequences B(n,2) (which has a known explicit formula) can be used to give the fairly tight lower bound that a(n) > 2*n^2*B(n,2). See A166315.