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

Terms

a(0) =1a(1) =2a(2) =6a(3) =21a(4) =78a(5) =297a(6) =1145a(7) =4447a(8) =17358a(9) =68001a(10) =267141a(11) =1051767a(12) =4148281a(13) =16385111a(14) =64797543a(15) =256515731a(16) =1016368078a(17) =4030114641a(18) =15990813773a(19) =63485616391a(20) =252175202373