Lattice points on ellipses

Topic:
Ellipse

How weird ?

These growing ellipses have all the Z² lattice points on their boundary. Move the slider k and observe the equation of the red ellipse. It's always x²-xy+y²=n, but not for every integer n...

I tried to reproduce a somewhat surprising result exposed in this awesome video by Edmund Harriss ans Pierre Arnoux. Next step is to make the points rotate.

Animated GIF extracted from the video.

Animated GIF extracted from the video.