a(n) = Sum_{k=0..floor(n/3)} Catalan(n-3*k).

Terms

a(0) =1a(1) =1a(2) =2a(3) =6a(4) =15a(5) =44a(6) =138a(7) =444a(8) =1474a(9) =5000a(10) =17240a(11) =60260a(12) =213012a(13) =760140a(14) =2734700a(15) =9907857a(16) =36117810a(17) =132379490a(18) =487546557a(19) =1803381000a(20) =6696499910a(21) =24953813577a(22) =93285944640a(23) =349756113560