What is the significance of the same leading term in a continued fraction expansion?

Question

The continued fraction expansion for the golden mean $\varphi = \frac{1+\sqrt{5}}{2}$ gives:

ContinuedFraction[(1 + Sqrt[5])/2, 50]: $$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, \ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, \ 1, 1, 1, 1, 1, 1, 1 $$

and the continued fraction expansion for the silver mean $\lambda = 1 + \sqrt{2}$ gives:

ContinuedFraction[1 + Sqrt[2], 50]: $$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, \ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, \ 2, 2, 2, 2 $$

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Is there a mathematical reason for the same leading term? \ Is there a way to predict which numbers will have a continued fraction expansion with the same leading coefficient?

Answer 1

see the following $$x=a+\frac{1}{a+\frac{1}{a+\frac{1}{..}}}$$ $$x=a+\frac{1}{x}$$ $$x^2=ax+1$$ $$x^2-ax-1=0$$ the last equation gives you the continued fraction with the same leading coefficient