a(n) = Sum_{k=0..floor(n/3)} 2^(n-3*k) * binomial(2*n-4*k+1,2*k).

Terms

a(0) =1a(1) =2a(2) =4a(3) =11a(4) =36a(5) =116a(6) =357a(7) =1078a(8) =3256a(9) =9879a(10) =30040a(11) =91352a(12) =277673a(13) =843802a(14) =2564124a(15) =7792163a(16) =23680412a(17) =71965100a(18) =218701901a(19) =664632878a(20) =2019811632a(21) =6138187055a(22) =18653894640a(23) =56689020016a(24) =172277420241a(25) =523549505522