Recently in my research I stumbled upon this splitting of a periodic continued fraction. I wondered whether there is any closed expression or literature on this topic.
Visualizing continued fractions by rectangles
It is known that a continued fraction $a = [n_0, n_1, ...]$ can be visualized by filling in squares of a rectangular sheet of paper (as described here). I illustrate the procedure on the periodic continued fraction $a = [\overline{3, 1, 2, 1}]$:
Take a rectangle with width $a$ and height $1$. Then we can fill in $3$ squares from the left, then $1$ square from the bottom, then $2$ squares from the left, then $1$ square from the bottom, then the procedure repeats.
Illustrating Rectangle filled by squares
Adapting the visualization to generate a real number
Instead of always filling the squares in from the left, I do the following:
- If $3$ squares are removed, then remove $2$ from the left and $1$ from the right.
- If $2$ squares are removed, then remove $1$ from the left and $1$ from the right.
When doing this, the yet "empty" space of the paper is no longer on the right but in the middle of the rectangle and converges to a specific x-coordinate.
Illustrating Left-Right-Fillings
Generalizing, this operation takes in a purely periodic continued fraction (here $a = [\overline{3, 1, 2, 1}]$) and a sequence of rational numbers (here $(\frac{2}{3}, \frac{1}{2})$) to split $a$ into two parts.
Is there any closed expression for the resulting x-coordinate? Is there literature on that kind of operation?