Solution to a task from IMO 1988 - calculation of examples

In 1988, the International Mathematical Olympiad was given the task B3: If for integers x and y the term (x^2+y^2)/(1+x*y) is a natural number N, then N is a square number. The solution does not require advanced mathematics, but is very challenging. I have attached a German language file MC14. I saved myself a translation into English because the solution should be self-explanatory with little translation help. Examples are included at the end of the file.

In the solution, a recursion is constructed based on starting values that have yet to be found. And that's my problem. My programming knowledge is not sufficient to find starting values and apply the recursion described in the solution - I therefore ask for help.