a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-k,n-3*k).

Terms

a(0) =1a(1) =2a(2) =6a(3) =21a(4) =77a(5) =288a(6) =1090a(7) =4159a(8) =15964a(9) =61557a(10) =238221a(11) =924597a(12) =3597290a(13) =14024341a(14) =54770176a(15) =214218966a(16) =838959762a(17) =3289471537a(18) =12910910288a(19) =50720828034a(20) =199422778415a(21) =784672001097