a(n) = a(n-1) + a(n-2) + 2 a(ceiling(n/2)), where a(0) = 1, a(1) = 1, a(2) = 1.

Terms

a(0) =1a(1) =1a(2) =1a(3) =4a(4) =7a(5) =19a(6) =34a(7) =67a(8) =115a(9) =220a(10) =373a(11) =661a(12) =1102a(13) =1897a(14) =3133a(15) =5260a(16) =8623a(17) =14323a(18) =23386a(19) =38455a(20) =62587a(21) =102364a(22) =166273a(23) =270841a(24) =439318a(25) =713953a(26) =1157065a(27) =1877284a(28) =3040615a(29) =4928419