a(n) = 2*a(n-1) - a(n-3) + a(ceiling(n/2)), where a(0) = 1, a(1) = 2, a(2) = 3.

Terms

a(0) =1a(1) =2a(2) =3a(3) =8a(4) =17a(5) =39a(6) =78a(7) =156a(8) =290a(9) =541a(10) =965a(11) =1718a(12) =2973a(13) =5137a(14) =8712a(15) =14741a(16) =24635a(17) =41099a(18) =67998a(19) =112326a(20) =184518a(21) =302756a(22) =494904a(23) =808263a(24) =1316743a(25) =2143719a(26) =3484312a(27) =5660593a(28) =9186179a(29) =14902787