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

Terms

a(0) =1a(1) =1a(2) =1a(3) =3a(4) =7a(5) =13a(6) =45a(7) =151a(8) =403a(9) =1617a(10) =6793a(11) =23275a(12) =105951a(13) =522133a(14) =2159077a(15) =10964223a(16) =61134955a(17) =293587801a(18) =1641566913a(19) =10124731987a(20) =55014334903a(21) =335177088285