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

Terms

a(0) =1a(1) =1a(2) =3a(3) =4a(4) =8a(5) =12a(6) =18a(7) =27a(8) =40a(9) =58a(10) =83a(11) =118a(12) =195a(13) =242a(14) =-1387a(15) =-338a(16) =75876a(17) =44491a(18) =-3099115a(19) =-2028539a(20) =129829195a(21) =91749709a(22) =-5687984421a(23) =-4236497556a(24) =263653557716a(25) =204087552038