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

Terms

a(0) =1a(1) =3a(2) =10a(3) =36a(4) =135a(5) =517a(6) =2003a(7) =7815a(8) =30634a(9) =120480a(10) =475002a(11) =1876294a(12) =7422676a(13) =29400192a(14) =116567356a(15) =462561572a(16) =1836843591a(17) =7298613997a(18) =29016050831a(19) =115408159467a(20) =459209330821