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

Terms

a(0) =1a(1) =1a(2) =2a(3) =6a(4) =19a(5) =62a(6) =208a(7) =713a(8) =2485a(9) =8776a(10) =31329a(11) =112848a(12) =409582a(13) =1496305a(14) =5497421a(15) =20298097a(16) =75276698a(17) =280264222a(18) =1047133901a(19) =3924785149a(20) =14753034485a(21) =55601758378a(22) =210059094449a(23) =795346129714