Lower (1/2)-midsequence of F(n) and F(n+4), where F = A000045 (Fibonacci numbers); see Comments.

Terms

a(0) =1a(1) =3a(2) =4a(3) =7a(4) =12a(5) =19a(6) =31a(7) =51a(8) =82a(9) =133a(10) =216a(11) =349a(12) =565a(13) =915a(14) =1480a(15) =2395a(16) =3876a(17) =6271a(18) =10147a(19) =16419a(20) =26566a(21) =42985a(22) =69552a(23) =112537a(24) =182089a(25) =294627a(26) =476716a(27) =771343a(28) =1248060a(29) =2019403