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

Terms

a(0) =1a(1) =3a(2) =10a(3) =36a(4) =132a(5) =490a(6) =1837a(7) =6939a(8) =26367a(9) =100673a(10) =385917a(11) =1484327a(12) =5725413a(13) =22138911a(14) =85790874a(15) =333080508a(16) =1295351352a(17) =5045196654a(18) =19676805486a(19) =76835246652a(20) =300363422658