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 0,2,3.

Terms

a(0) =0a(1) =2a(2) =3a(3) =5a(4) =8a(5) =15a(6) =28a(7) =53a(8) =99a(9) =185a(10) =345a(11) =644a(12) =1202a(13) =2244a(14) =4189a(15) =7820a(16) =14598a(17) =27251a(18) =50871a(19) =94964a(20) =177275a(21) =330930a(22) =617767a(23) =1153223a(24) =2152791a(25) =4018745a(26) =7502034a(27) =14004500a(28) =26143046a(29) =48802803