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

Terms

a(0) =1a(1) =1a(2) =1a(3) =3a(4) =13a(5) =37a(6) =87a(7) =241a(8) =793a(9) =2513a(10) =7437a(11) =22287a(12) =70051a(13) =222883a(14) =700213a(15) =2195139a(16) =6959869a(17) =22252933a(18) =71201129a(19) =227826699a(20) =731309001a(21) =2356460041a(22) =7609531843a(23) =24603325189a(24) =79677148959a(25) =258535824775a(26) =840291483835