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 2,2,3.

Terms

a(0) =2a(1) =2a(2) =3a(3) =5a(4) =10a(5) =19a(6) =36a(7) =67a(8) =125a(9) =233a(10) =435a(11) =812a(12) =1516a(13) =2830a(14) =5283a(15) =9862a(16) =18410a(17) =34367a(18) =64155a(19) =119762a(20) =223567a(21) =417346a(22) =779085a(23) =1454365a(24) =2714951a(25) =5068163a(26) =9461046a(27) =17661506a(28) =32969800a(29) =61546717