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 0,1,4.

Terms

a(0) =0a(1) =1a(2) =4a(3) =5a(4) =9a(5) =15a(6) =29a(7) =54a(8) =102a(9) =190a(10) =355a(11) =662a(12) =1236a(13) =2307a(14) =4307a(15) =8040a(16) =15009a(17) =28018a(18) =52303a(19) =97637a(20) =182265a(21) =340245a(22) =635156a(23) =1185684a(24) =2213388a(25) =4131865a(26) =7713202a(27) =14398700a(28) =26878923a(29) =50176509