This representations of the repeated iteration of should be downloaded. The points are colored based upon where they are located when k=1. You can also reveal the orbit of the points by using the step slider. Move point A to change the values of the first two numbers in the sequence. Drag the slider k to change what is being added in the numerator.
The invariant curve has the equation where C in a constant.
There are up to 200 points displayed. When you download the files, show the Spreadsheet View and add more points by filling more points down.

Does there appear to be other orbits with periods different from 5?

What kinds of symmetry do you notice?

Does there appear to be any "special" values of k?

Does there appear to be any special locations for point A?