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

Terms

a(0) =1a(1) =1a(2) =3a(3) =5a(4) =11a(5) =22a(6) =44a(7) =92a(8) =181a(9) =379a(10) =750a(11) =1557a(12) =3109a(13) =6401a(14) =12872a(15) =26349a(16) =53228a(17) =108575a(18) =219918a(19) =447702a(20) =908140a(21) =1846804a(22) =3749001a(23) =7619870a(24) =15474230a(25) =31443013a(26) =63865592a(27) =129755543a(28) =263576791a(29) =535475968