A self-"read and extend" sequence built following the rules given in the Comments section.

Terms

a(0) =1a(1) =2a(2) =4a(3) =8a(4) =16a(5) =15a(6) =30a(7) =29a(8) =58a(9) =57a(10) =56a(11) =112a(12) =224a(13) =223a(14) =446a(15) =445a(16) =890a(17) =889a(18) =1778a(19) =1777a(20) =3a(21) =6a(22) =5a(23) =7a(24) =14a(25) =13a(26) =9a(27) =18a(28) =36a(29) =10