a(n) = Sum_{k=0..floor(n/4)} binomial(n+2*k,n-4*k).

Terms

a(0) =1a(1) =1a(2) =1a(3) =1a(4) =2a(5) =8a(6) =29a(7) =85a(8) =212a(9) =476a(10) =1016a(11) =2172a(12) =4825a(13) =11213a(14) =26763a(15) =64095a(16) =151851a(17) =354737a(18) =820328a(19) =1889968a(20) =4361521a(21) =10106859a(22) =23509678a(23) =54793282a(24) =127709888a(25) =297336790a(26) =691382201a(27) =1606284377a(28) =3731020629a(29) =8668253125