a(n) = Sum_{k=0..floor(n/3)} 2^(n-3*k) * binomial(2*n-4*k+1,2*k+1).

Terms

a(0) =1a(1) =6a(2) =20a(3) =57a(4) =164a(5) =492a(6) =1505a(7) =4602a(8) =14008a(9) =42545a(10) =129208a(11) =392584a(12) =1193121a(13) =3626142a(14) =11020076a(15) =33489769a(16) =101774332a(17) =309290388a(18) =939930945a(19) =2856443826a(20) =8680709552a(21) =26380596577a(22) =80170385200a(23) =243637058512a(24) =740410793665