a(n) = ((15*n + 34)/54)*2^(n-1) - (-1)^(n-1)*(6*n + 5)/27.

Terms

a(0) =3a(1) =5a(2) =15a(3) =31a(4) =75a(5) =163a(6) =367a(7) =799a(8) =1747a(9) =3771a(10) =8119a(11) =17367a(12) =37019a(13) =78579a(14) =166271a(15) =350735a(16) =737891a(17) =1548587a(18) =3242823a(19) =6776903a(20) =14136363a(21) =29437795a(22) =61205775a(23) =127071871a(24) =263464435a(25) =545570203a(26) =1128423127a(27) =2331411639a(28) =4811954107