a(n+4) = a(n+3)+a(n+1)+a(n)+k(n), where k(n) = 0, 1, 0, or -1 according to n mod 4.

Terms

a(0) =0a(1) =0a(2) =1a(3) =1a(4) =1a(5) =3a(6) =5a(7) =6a(8) =10a(9) =19a(10) =30a(11) =45a(12) =74a(13) =124a(14) =199a(15) =317a(16) =515a(17) =839a(18) =1355a(19) =2186a(20) =3540a(21) =5735a(22) =9276a(23) =15001a(24) =24276a(25) =39288a(26) =63565a(27) =102841a(28) =166405a(29) =269259