f(f(n+1))-f(f(n)), where f(m) = binary partition(m) = A000123(m).

Terms

a(0) =2a(1) =6a(2) =10a(3) =40a(4) =80a(5) =250a(6) =510a(7) =1890a(8) =4270a(9) =13738a(10) =30630a(11) =101960a(12) =234864a(13) =705046a(14) =1580578a(15) =5136840a(16) =11991928a(17) =36095322a(18) =82990606a(19) =255579576a(20) =595920680a(21) =1719038038a(22) =3926998938a(23) =11665004238a(24) =26978093154a(25) =75938910058a(26) =172678519950a(27) =489262646480