I guess you're a little wrong with your few values, for example, I get
Hee, I got a really simple recurrence relation:
Calculating a few values we easily notice ,
but how is this formally derived?
Can someone offer a quick insight?
... is non-linear and in most cases like that an ad hoc solving procedure has to be found. In this particular case it is easy to see that the solution is a continued fraction...
... where . Now if You use the standard algorithm to write the rational number in term of continued fraction You obtain exactly the expression (2) so that is...