a(n) = max_{r=1..n-1} ceiling(t(t(n)-t(r-1))/(n-r+1)), where t() = triangular numbers A000217.

Terms

a(0) =1a(1) =3a(2) =8a(3) =15a(4) =27a(5) =43a(6) =65a(7) =94a(8) =130a(9) =175a(10) =229a(11) =294a(12) =369a(13) =456a(14) =557a(15) =671a(16) =800a(17) =944a(18) =1105a(19) =1283a(20) =1479a(21) =1695a(22) =1930a(23) =2187a(24) =2465a(25) =2765a(26) =3090a(27) =3439a(28) =3813a(29) =4213