[math-fun] The 3 terms after "3" cannot be 4, 5 or 6.

Hello Math-fun, (thanks for the private answers to my earlier posts on other topics) --- Start the sequence S with a(1)=1. The next term is the smallest one not yet present in S with the rule: the a(n) terms immediately after a(n) must be < a(n) or > 2*a(n). Example: The 7 terms immediately after 7 cannot be 8, 9, 10, ... 14. The 15 termrs immediately after 15 cannot be 16, 17, 18, ..., 30. ... The term immediately after 1 cannot be 2. S starts like this (I think): s = 1, 3, 2, 7, 15, 4, 31, 63, 127, 255, 5, 8, ... Could please someone interested compute a hundred or so terms and find a formula giving the n-th term? I hope this is not old hat, Best, É.