Let u(1)=u(2)=u(3)=2, u(n)=(1+u(n-1)u(n-2))/u(n-3); then a(n) is the numerator of u(n).

Terms

a(0) =2a(1) =2a(2) =2a(3) =5a(4) =3a(5) =17a(6) =11a(7) =65a(8) =43a(9) =257a(10) =171a(11) =1025a(12) =683a(13) =4097a(14) =2731a(15) =16385a(16) =10923a(17) =65537a(18) =43691a(19) =262145a(20) =174763a(21) =1048577a(22) =699051a(23) =4194305a(24) =2796203a(25) =16777217a(26) =11184811a(27) =67108865a(28) =44739243a(29) =268435457