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

Terms

a(0) =1a(1) =4a(2) =15a(3) =57a(4) =220a(5) =858a(6) =3368a(7) =13276a(8) =52479a(9) =207861a(10) =824527a(11) =3274395a(12) =13015081a(13) =51769813a(14) =206045841a(15) =820475513a(16) =3268499356a(17) =13025237058a(18) =51922543076a(19) =207034128448a(20) =825713206746