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 1,1,4.

Terms

a(0) =1a(1) =1a(2) =4a(3) =5a(4) =10a(5) =17a(6) =33a(7) =61a(8) =115a(9) =214a(10) =400a(11) =746a(12) =1393a(13) =2600a(14) =4854a(15) =9061a(16) =16915a(17) =31576a(18) =58945a(19) =110036a(20) =205411a(21) =383453a(22) =715815a(23) =1336255a(24) =2494468a(25) =4656574a(26) =8692708a(27) =16227203a(28) =30292300a(29) =56548466