Defined in comments.

Terms

a(0) =3a(1) =9a(2) =31a(3) =106a(4) =365a(5) =1263a(6) =4388a(7) =15336a(8) =53871a(9) =190059a(10) =673222a(11) =2393291a(12) =8535397a(13) =30526712a(14) =109449848a(15) =393272258a(16) =1415768769a(17) =5105086517a(18) =18434398665a(19) =66647658995a(20) =241210652738a(21) =873773659486