User blog comment:QuasarBooster/Fibonacci/Lucas Sequence Extentions?/@comment-2033667-20150730151541/@comment-1605058-20150730195910

If you have recurrence S_{n+k}=a_{k-1}S_{n+k-1}+...+a_1S_{n+1}+a_0S_n then, if polynomial $$x^k-a_{k-1}x^{k-1}+...+a_1x+a_0$$ has root $$\alpha$$ is a complex root of absolute value greater than other roots, then I'm fairly sure the ratio tends to $$\alpha$$. I'm not sure what happens when $$\alpha$$ is a repeated root or there are many roots with maximal absolute value.