a(n) = Sum_{k=0..n} binomial(n, floor(k/2))*3^(n-k).

Terms

a(0) =1a(1) =4a(2) =14a(3) =48a(4) =162a(5) =544a(6) =1820a(7) =6080a(8) =20290a(9) =67680a(10) =225684a(11) =752448a(12) =2508468a(13) =8362176a(14) =27875064a(15) =92919168a(16) =309734850a(17) =1032458080a(18) =3441543140a(19) =11471842880a(20) =38239537852a(21) =127465249344a(22) =424884399624