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

Terms

a(0) =1a(1) =2a(2) =4a(3) =8a(4) =16a(5) =32a(6) =46a(7) =92a(8) =841a(9) =1268a(10) =2356a(11) =1247a(12) =9442a(13) =14888a(14) =97762a(15) =955421a(16) =554219a(17) =113488a(18) =226679a(19) =334558a(20) =966611a(21) =1222339a(22) =2444678a(23) =3456889a(24) =9877631a(25) =12255679a(26) =11234558a(27) =11224669a(28) =22334489a(29) =98876644