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

Terms

a(0) =1a(1) =1a(2) =3a(3) =11a(4) =40a(5) =147a(6) =547a(7) =2055a(8) =7777a(9) =29602a(10) =113204a(11) =434591a(12) =1673821a(13) =6464539a(14) =25026534a(15) =97087873a(16) =377329971a(17) =1468856383a(18) =5726159811a(19) =22351657810a(20) =87350137071a(21) =341726039806