a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(2*n-k+1,n-3*k).

Terms

a(0) =1a(1) =3a(2) =10a(3) =34a(4) =118a(5) =417a(6) =1497a(7) =5447a(8) =20047a(9) =74493a(10) =279054a(11) =1052467a(12) =3992204a(13) =15216662a(14) =58239175a(15) =223688159a(16) =861769598a(17) =3328779906a(18) =12887832493a(19) =49998248601a(20) =194315972151a(21) =756406944446