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

Terms

a(0) =1a(1) =1a(2) =1a(3) =2a(4) =3a(5) =4a(6) =7a(7) =12a(8) =19a(9) =34a(10) =63a(11) =112a(12) =211a(13) =414a(14) =799a(15) =1588a(16) =3267a(17) =6706a(18) =13999a(19) =30024a(20) =64723a(21) =141142a(22) =314271a(23) =705724a(24) =1599619a(25) =3685338a(26) =8573167a(27) =20112016a(28) =47804499a(29) =114743614