Lower (2/3)-midsequence of A000032 (Lucas numbers) and A000045 (Fibonacci numbers); see Comments.

Terms

a(0) =1a(1) =1a(2) =2a(3) =4a(4) =6a(5) =10a(6) =17a(7) =28a(8) =45a(9) =73a(10) =118a(11) =192a(12) =310a(13) =502a(14) =813a(15) =1316a(16) =2129a(17) =3445a(18) =5574a(19) =9020a(20) =14594a(21) =23614a(22) =38209a(23) =61824a(24) =100033a(25) =161857a(26) =261890a(27) =423748a(28) =685638a(29) =1109386