a(1) = 1, a(n) = Sum_{k=1..n} (k mod 3) * a(n-k) for n >= 2.

Terms

a(0) =1a(1) =1a(2) =3a(3) =5a(4) =12a(5) =25a(6) =54a(7) =116a(8) =249a(9) =535a(10) =1149a(11) =2468a(12) =5301a(13) =11386a(14) =24456a(15) =52529a(16) =112827a(17) =242341a(18) =520524a(19) =1118033a(20) =2401422a(21) =5158012a(22) =11078889a(23) =23796335a(24) =51112125a(25) =109783684a(26) =235804269a(27) =506483762a(28) =1087875984a(29) =2336647777