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

Terms

a(0) =1a(1) =2a(2) =6a(3) =19a(4) =64a(5) =224a(6) =805a(7) =2947a(8) =10934a(9) =40975a(10) =154738a(11) =587910a(12) =2244681a(13) =8605061a(14) =33099767a(15) =127687258a(16) =493796454a(17) =1913755319a(18) =7431027611a(19) =28902878561a(20) =112585961052a(21) =439148770623