a(n) = Sum_{k=0..n} binomial(n, floor(k/2))*4^(n-k).

Terms

a(0) =1a(1) =5a(2) =22a(3) =95a(4) =406a(5) =1730a(6) =7360a(7) =31295a(8) =133030a(9) =565430a(10) =2403172a(11) =10213670a(12) =43408444a(13) =184486580a(14) =784069252a(15) =3332296895a(16) =14162266630a(17) =60189642830a(18) =255806000260