Is anything known about continued fractions in which the sequence of integers is quasiperiodic?

Quasiperiodic is meant here in the sense of 1D quasicrystals. For example, draw an irrationally-sloped straight line through a 2D lattice: whenever the line intersects a vertical line of the lattice write integer $A$, and whenever the line intersects a horizontal line of the lattice write (different) integer $B$.

For example, one sequence generated by slope $(1-\sqrt{5})/2$ with $A=1,\,B=2$ is

$1,\,2,\, 2,\, 1,\, 2,\, 2,\, 1,\, 2,\, 1,\, 2,\, 2,\, 1,\, 2,\, 1,\, 2,\, 2,\, 1,\, 2,\, 2,\, 1,\, \ldots$

in which case the expansion (integer part zero) evaluates to

$0.70326125114456817\ldots$

My understanding is that the continued fraction expansion of a 'generic' irrational number is expected to comprise numbers approximately following the Gauss-Kuzmin distribution

$p(k)=-\log_2\left(1-\left(1+k\right)^{-2}\right).$

In this sense, quadratic irrationals are non-generic, having periodic continued fraction expansions. I'm wondering if the quasiperiodic case might be close enough to periodic that it's covered by some results applying to quadratic irrationals but not to generic irrationals.

Any thoughts would be greatly appreciated!