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 1,1,3.

Terms

a(0) =1a(1) =1a(2) =3a(3) =4a(4) =8a(5) =14a(6) =27a(7) =50a(8) =94a(9) =175a(10) =327a(11) =610a(12) =1139a(13) =2126a(14) =3969a(15) =7409a(16) =13831a(17) =25819a(18) =48198a(19) =89974a(20) =167960a(21) =313541a(22) =585306a(23) =1092626a(24) =2039671a(25) =3807577a(26) =7107834a(27) =13268623a(28) =24769340a(29) =46238423