This representations of the repeated iteration of [math]a_{n+2} = \frac{a_{n+1}+k}{a_n}[/math] 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[math]\left(a_1,a_2 \right)[/math] 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. [br][br]The invariant curve has the equation [math]\frac{(x+1)(y+1)(x+y+k)}{xy}=C[/math] where C in a constant.[br][br]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.

[list][*]Does there appear to be other orbits with periods different from 5? [br][/*][*]What kinds of symmetry do you notice? [br][/*][*]Does there appear to be any "special" values of k? [br][/*][*]Does there appear to be any special locations for point A?[br][/*][/list]