Here is a picture of "square numbers" from mathworld website.
A plot of the first few square numbers represented as a sequence of binary bits is shown above. The top portion shows $1^2$ to $255^2$, and the bottom shows the next 510 values. (Black is for "one" bit, and while is for "zero" bit)
I want to learn more about this pattern. However, there is no further explanation about the pattern of this plot from the website.
Is there any known result about this pattern ?