If the binary representation of the binary string 10s00 is in the sequence, so is 101s000.
For decimal representation see A306514.
This sequence is a subset of A066059.
These regular patterns can be represented by the context-free grammar with production rules:
S -> S_a | S_b | S_c | S_d
S_a -> 10 T_a 00, T_a -> 1 T_a 0 | T_a0,
S_b -> 11 T_b 01, T_b -> 0 T_b 1 | T_b0,
S_c -> 10 T_c 000, T_c -> 1 T_c 0 | T_c0,
S_d -> 11 T_d 101, T_d -> 0 T_d 1 | T_d0,
where T_a0, T_b0, T_c0 and T_d0 are some terminating strings.
Numbers in this sequence are generated by choosing S_a or S_c from the starting symbol S.
From the fact that all strings derived from S_b have prefix 11 and suffix 00, it can be proved that all strings derived from S_a must have prefix 111 (i.e., 1 is prefix of s, with s as in the name of this sequence). Similarly, from the fact that all strings derived from S_d have prefix 11 and suffix 000, it can be proved that all strings derived from S_c must have prefix 111 (i.e., again, 1 is prefix of s, with s as in the name of this sequence). In the later case, 11 is a prefix of s, which is even stronger. I believe additional stronger conditions can be observed and proved, so I challenge others to take a look at it too.