a(n) = b_f(n) where f is the 2-periodic sequence f(k) = (-1)^k (see comments).

Terms

a(0) =2a(1) =5a(2) =11a(3) =23a(4) =41a(5) =61a(6) =107a(7) =197a(8) =311a(9) =617a(10) =1229a(11) =2381a(12) =4649a(13) =8861a(14) =17027a(15) =33809a(16) =67409a(17) =134681a(18) =267719a(19) =535349a(20) =1069217a(21) =2138399a(22) =4275641a(23) =8545697a(24) =17091377a(25) =34182749a(26) =68365469a(27) =136730639a(28) =273461159a(29) =546917141