a(n) = Sum{a(k): k=0,1,2,...,n-4,n-2,n-1}; a(n-3) is not a summand; initial terms are 1,3,3.

Terms

a(0) =1a(1) =3a(2) =3a(3) =6a(4) =10a(5) =20a(6) =37a(7) =70a(8) =130a(9) =243a(10) =453a(11) =846a(12) =1579a(13) =2948a(14) =5503a(15) =10273a(16) =19177a(17) =35799a(18) =66828a(19) =124752a(20) =232882a(21) =434735a(22) =811546a(23) =1514962a(24) =2828071a(25) =5279331a(26) =9855246a(27) =18397383a(28) =34343506a(29) =64111097