a(n) = Sum_{k=0..floor(n/2)} 2^((n-2*k)*k) for n>=0.

Terms

a(0) =1a(1) =1a(2) =2a(3) =3a(4) =6a(5) =13a(6) =34a(7) =105a(8) =386a(9) =1681a(10) =8706a(11) =53793a(12) =395266a(13) =3442753a(14) =35659778a(15) =440672385a(16) =6476038146a(17) =112812130561