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

Terms

a(0) =2a(1) =2a(2) =3a(3) =4a(4) =7a(5) =11a(6) =18a(7) =28a(8) =46a(9) =74a(10) =119a(11) =192a(12) =311a(13) =503a(14) =814a(15) =1316a(16) =2130a(17) =3446a(18) =5575a(19) =9020a(20) =14595a(21) =23615a(22) =38210a(23) =61824a(24) =100034a(25) =161858a(26) =261891a(27) =423748a(28) =685639a(29) =1109387