The usual way of analyzing three-term recurrences is to look at the asymptotic behavior of your $F(a(n))$ and $G(a(n))$ and then invoke the machinery of (Poincaré-)Perron. What often happens is that there is one solution that is decaying (the "minimal" solution) and one that is growing (the "dominant" solution).

Usually people are interested in the bounded or minimal solution, and attempting to propagate the recurrence forward (increasing n) in finite precision is courting certain numerical disaster. Due to rounding, the initial values of your recurrence can be expressed as a linear combination of the minimal and dominant solution; thus, if the recurrence is executed forward, there will come a point where the growth of the dominant solution will swamp the behavior of the bounded solution. (In numerical work, people usually recurse backwards or use special techniques like Miller's method the Miller algorithm when they are interested in minimal solutions)

Related to this, it has long been known that certain multistep methods for solving ODEs (e.g. Milne's method) can exhibit the phenomenon of "parasitic solutions", where no matter how small a step size $h$ you take, a growing solution to the recurrence will alternate that alternates in sign with every step , can easily contaminating contaminate your bounded solutionand , rendering your numbers meaningless.

At this point, I would like to direct you to this excellent survey article by Walter Gautschi and this old but nice book by Jet Wimp.

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The usual way of analyzing three-term recurrences is to look at the asymptotic behavior of your $F(a(n))$ and $G(a(n))$ and then invoke the machinery of (Poincaré-)Perron. What often happens is that there is one solution that is decaying (the "minimal" solution) and one that is growing (the "dominant" solution).

Usually people are interested in the bounded or minimal solution, and attempting to propagate the recurrence forward (increasing n) in finite precision is courting certain numerical disaster. Due to rounding, the initial values of your recurrence can be expressed as a linear combination of the minimal solution; thus, if the recurrence is executed forward, there will come a point where the growth of the dominant solution will swamp the behavior of the bounded solution. (In numerical work, people usually recurse backwards or use special techniques like Miller's method when they are interested in minimal solutions)

Related to this, it has long been known that certain multistep methods for solving ODEs (e.g. Milne's method) can exhibit the phenomenon of "parasitic solutions", where no matter how small a step size $h$ you take, a growing solution to the recurrence will alternate in sign with every step, easily contaminating your bounded solution and rendering your numbers meaningless.

At this point, I would like to direct you to this excellent survey article by Walter Gautschi and this old but nice book by Jet Wimp.