Hidden patterns in the natural numbers

Solution 1:

1. At the heart of the patterns is the identity $n^2 - k^2 \equiv 0 \mod (n\pm k)$. The right-most dots $a_{ij} = 1$ on the wave fronts along the orange line have $i = j^2$. The right-most dots on the wave fronts along the other directions have $i = 2\,j^2$ (blue) and $2\,i=j^2$ (green).

2. Giving those dots $a_{ij}$ with $i = n^2 - k^2$, $j = n\pm k$ the color $n$ makes the wave fronts distinguished and reveal that they are parabolas:
   Since natural numbers may be written in zero to many ways as the difference of two squares, some $a_{ij}$ lie on no and some lie on several parabolas.

3. The orange line itself is in fact a parabola rotated by 90 degrees.