Consider binary words that begin with 1 such that the subword 00, whenever it appears, is followed by 111. Then a(n) counts such words at length n (including those where the string 111 is yet being completed - see Example).

A340217

Consider binary words that begin with 1 such that the subword 00, whenever it appears, is followed by 111. Then a(n) counts such words at length n (including those where the string 111 is yet being completed - see Example).