a(n)=a(n-1)+a(n-2)-d, where d=a(n/2) if n is even, else d=0; 2 initial terms.

Terms

a(0) =1a(1) =0a(2) =1a(3) =1a(4) =2a(5) =2a(6) =4a(7) =5a(8) =9a(9) =12a(10) =21a(11) =31a(12) =52a(13) =79a(14) =131a(15) =205a(16) =336a(17) =532a(18) =868a(19) =1388a(20) =2256a(21) =3623a(22) =5879a(23) =9471a(24) =15350a(25) =24769a(26) =40119a(27) =64809a(28) =104928a(29) =169606