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

Terms

a(0) =1a(1) =4a(2) =15a(3) =57a(4) =219a(5) =847a(6) =3290a(7) =12819a(8) =50066a(9) =195909a(10) =767790a(11) =3013002a(12) =11837043a(13) =46548919a(14) =183209125a(15) =721628692a(16) =2844297119a(17) =11217639757a(18) =44265835891a(19) =174765349896a(20) =690308413773