What is the pattern?

What is the pattern in the progression of ratios for rectangles with features similar to the golden rectangle?

In my previous blogs on rectangles I showed that the golden rectangle is not so distinctive in its feature that when a square section is removed, the remainder is another rectangle with the same proportions as the first. I described rectangles that when 2, 3, 4, 5, etcetera, square sections are removed, the remainder is another rectangle with the same proportions as the first.

Here is the pattern of ratios for this series.

(1+√5)/2 [The golden ratio]

1+√2

(3+√13)/2

2+√5

(5+√29)/2

3+√10

(7+√53)/2

4+√17

(9+√85)/2

5+√26

(5+5*√5)/2

6+√37

These are the ratios of long side to short side for rectangles that, starting with the golden rectangle, are in proportion such that if one, two, three, etcetera, unit rectangles are removed, the remaining rectangle is in the same proportion as the original. What's going on here?