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

Terms

a(0) =1a(1) =2a(2) =3a(3) =6a(4) =15a(5) =36a(6) =80a(7) =172a(8) =369a(9) =796a(10) =1727a(11) =3774a(12) =8322a(13) =18528a(14) =41643a(15) =94460a(16) =216121a(17) =498186a(18) =1155147a(19) =2689626a(20) =6278841a(21) =14676900a(22) =34316598a(23) =80194032a(24) =187195554a(25) =436310190a(26) =1015176726a(27) =2357708258a(28) =5465611759a(29) =12647864454