This is just for curiosity. I noticed, that certain multiples of square rooted numbers can be formed with a periodic continued fraction. Addition to that periodicity is reflective by nature. Say for example:
$$9 \sqrt3 = [15; 1, 1, 2, 3, 15, 3, 2, 1, 1, 30]$$
Where:
- start element of the continued fraction: $15$
- repetitive period is: $1, 1, 2, 3, 15, 3, 2, 1, 1, 30$
- middle element is: $15$
- reflective parts are: $1, 1, 2, 3$ and $3, 2, 1, 1$
- the last element of the period is the double of the start element: $30$
Other example:
$$16 \sqrt2 = [22; 1, 1, 1, 2, 6, 11, 6, 2, 1, 1, 1, 44]$$
It looks like this sort of sequences occur with (square?) rooted (prime?) numbers multiplied with powered? (prime?) numbers.
I want to know, if this behavior is identified somehow and if following the above five step procedure can produce a fixed set of (irrational?) numbers.
The most curious I'm about the consequence of the reflection of the number sequence. Is there an explanation for it?