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 1,2,4.

Terms

a(0) =1a(1) =2a(2) =4a(3) =6a(4) =11a(5) =20a(6) =38a(7) =71a(8) =133a(9) =248a(10) =463a(11) =864a(12) =1613a(13) =3011a(14) =5621a(15) =10493a(16) =19588a(17) =36566a(18) =68260a(19) =127425a(20) =237872a(21) =444050a(22) =828935a(23) =1547423a(24) =2888668a(25) =5392451a(26) =10066414a(27) =18791583a(28) =35079383a(29) =65484803