a(n) = L(n) + 2^n where L(n) = A000032(n) (the Lucas numbers).

Terms

a(0) =3a(1) =3a(2) =7a(3) =12a(4) =23a(5) =43a(6) =82a(7) =157a(8) =303a(9) =588a(10) =1147a(11) =2247a(12) =4418a(13) =8713a(14) =17227a(15) =34132a(16) =67743a(17) =134643a(18) =267922a(19) =533637a(20) =1063703a(21) =2121628a(22) =4233907a(23) =8452687a(24) =16880898a(25) =33722193a(26) =67380307a(27) =134656932a(28) =269146103a(29) =538020763