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

Terms

a(0) =1a(1) =1a(2) =1a(3) =1a(4) =2a(5) =3a(6) =4a(7) =5a(8) =8a(9) =13a(10) =20a(11) =29a(12) =46a(13) =77a(14) =128a(15) =205a(16) =338a(17) =581a(18) =1012a(19) =1733a(20) =2990a(21) =5293a(22) =9536a(23) =17117a(24) =30778a(25) =56165a(26) =104108a(27) =193621a(28) =360662a(29) =677693