a(n) = a(n-1) + a(n-2) + 2 a(ceiling(n/2)), where a(0) = 1, a(1) = 2, a(2) = 3.

Terms

a(0) =1a(1) =2a(2) =3a(3) =11a(4) =20a(5) =53a(6) =95a(7) =188a(8) =323a(9) =617a(10) =1046a(11) =1853a(12) =3089a(13) =5318a(14) =8783a(15) =14747a(16) =24176a(17) =40157a(18) =65567a(19) =107816a(20) =175475a(21) =286997a(22) =466178a(23) =759353a(24) =1231709a(25) =2001698a(26) =3244043a(27) =5263307a(28) =8524916a(29) =13817717