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 2,3,3.

Terms

a(0) =2a(1) =3a(2) =3a(3) =6a(4) =11a(5) =22a(6) =41a(7) =77a(8) =143a(9) =267a(10) =498a(11) =930a(12) =1736a(13) =3241a(14) =6050a(15) =11294a(16) =21083a(17) =39357a(18) =73470a(19) =137151a(20) =256028a(21) =477943a(22) =892205a(23) =1665533a(24) =3109151a(25) =5804040a(26) =10834752a(27) =20225886a(28) =37756883a(29) =70483054