a(n) = Sum_{k=0..floor(n/2)} T(n-k,k), T given by A026780.

Terms

a(0) =1a(1) =1a(2) =2a(3) =4a(4) =7a(5) =12a(6) =23a(7) =41a(8) =72a(9) =135a(10) =243a(11) =432a(12) =804a(13) =1455a(14) =2608a(15) =4836a(16) =8785a(17) =15838a(18) =29306a(19) =53385a(20) =96654a(21) =178600a(22) =326019a(23) =592140a(24) =1093135a(25) =1998537a(26) =3638700a(27) =6712659a(28) =12287071a(29) =22412784