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

Terms

a(0) =1a(1) =1a(2) =1a(3) =5a(4) =11a(5) =19a(6) =57a(7) =149a(8) =325a(9) =841a(10) =2223a(11) =5387a(12) =13599a(13) =35435a(14) =89561a(15) =227397a(16) =587861a(17) =1508833a(18) =3865285a(19) =9980185a(20) =25767813a(21) =66439737a(22) =171835439a(23) =445009955a(24) =1152176511a(25) =2986869731a(26) =7752069847a(27) =20125928723a(28) =52286212535a(29) =135949102339