a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, floor((n-k)/2)).

Terms

a(0) =1a(1) =1a(2) =3a(3) =5a(4) =11a(5) =19a(6) =39a(7) =71a(8) =141a(9) =261a(10) =513a(11) =965a(12) =1889a(13) =3585a(14) =7017a(15) =13417a(16) =26287a(17) =50527a(18) =99147a(19) =191399a(20) =376155a(21) =728619a(22) =1434051a(23) =2785667a(24) =5489823a(25) =10689199a(26) =21089799a(27) =41146383a(28) =81262983a(29) =158818311