a(n) = Sum_{k=0..floor(n/4)} n^k * binomial(n-3*k,k).

Terms

a(0) =1a(1) =1a(2) =1a(3) =1a(4) =5a(5) =11a(6) =19a(7) =29a(8) =105a(9) =298a(10) =671a(11) =1299a(12) =3997a(13) =12468a(14) =33083a(15) =75781a(16) =220625a(17) =708867a(18) =2086183a(19) =5412778a(20) =15756741a(21) =51093316a(22) =160523859a(23) =457283931a(24) =1365001273a(25) =4458076176a(26) =14608351135a(27) =44649287452a(28) =137979763181a(29) =455582050840