Does this sequence even exist?

This is from IMO 2009 Short List and is quite hard to solve.

The sequence exists for $n=1,2,3,4$ .For $n=1,2,3$ , a sequence can be easily constructed.But for $n=4$ , its a bit tough to construct a sequence.For $n=4$ ,a sequence can be $a_{1}=4,a_{2}=33,a_{3}=217$ and $a_{4}=1384$ .

I will now give a brief sketch of the solution:

Proving that there exist no sequence for $n=5$ is enough.The proof is by contradiction. Assume such a sequence exist.

At first,prove that $a_{1}$ cannot be odd,which will be comparatively easy to show.

So, assume $a_{1}$ is even.Now,you have to show it is also not possible,which is a bit tough.In this case,at first,show that $a_{2}$ cannot be odd.(which is easy).

So,then you can assume $a_{2}$ is even.One have to show that it is also impossible.

See that if $a_{2}$ is even,then $a_{4}$ and $a_{5}$ are also even. Note that $a_{2}+1|a_{3}^{2}+1$ and $a_{3}+1|a_{2}^{2}+1$ .Now, you have to show a more general statement:"there exist no even $x$ and $y$ such that $x+1|y^{2}+1$ and $y+1|x^{2}+1$ "To do this first show, both $(x+1)$ and $(y+1)$ divides $(x^{2}+y^{2})$ .Also show that $gcd(x+1,y+1)=1$ . So,you can find integer $k$ such that $k(x+1)(y+1)=x^{2}+y^{2}$ . Now,use Vieta root jumping to show this equation has no solution.