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

Terms

a(0) =1a(1) =1a(2) =1a(3) =4a(4) =17a(5) =46a(6) =112a(7) =365a(8) =1297a(9) =4126a(10) =12686a(11) =41647a(12) =141440a(13) =470887a(14) =1553021a(15) =5205769a(16) =17665105a(17) =59858038a(18) =202599814a(19) =689183087a(20) =2355887902a(21) =8065291637a(22) =27637715887a(23) =94924591313a(24) =326810899744