a(n) = Sum_{k>=1} floor(tau^(n-k)) where tau is A001622.

Terms

a(0) =1a(1) =2a(2) =4a(3) =8a(4) =14a(5) =25a(6) =42a(7) =71a(8) =117a(9) =193a(10) =315a(11) =514a(12) =835a(13) =1356a(14) =2198a(15) =3562a(16) =5768a(17) =9339a(18) =15116a(19) =24465a(20) =39591a(21) =64067a(22) =103669a(23) =167748a(24) =271429a(25) =439190a(26) =710632a(27) =1149836a(28) =1860482a(29) =3010333