a(n)=sum{k=0..n, C(n,floor(k/2))*(-1)^k*4^(n-k)}.

Terms

a(0) =1a(1) =3a(2) =14a(3) =57a(4) =246a(5) =1038a(6) =4424a(7) =18777a(8) =79846a(9) =339258a(10) =1442004a(11) =6128202a(12) =26045436a(13) =110691948a(14) =470442924a(15) =1999378137a(16) =8497365126a(17) =36113785698a(18) =153483619604a(19) =652305322542