Pixel coloring problem

We have an images $$m$$ pixels length and $$n$$ pixels width. There are at least $$k$$ pixels with different colors. Image and its reverse, reflect and rotation are considered to be equivalent. Permuting the colors will not change the structure. We select one concrete pixel and start to change its color, before all pixels in image will be same colored. Next we do the same for each other pixels, so for each pixel exist shortest way to make this image one-colored. There are two different pixels in any image, which have biggest and smallest values of shortest way (of course they may be equal). If $$m=1$$ and $$n,k$$ variable (or $$n=1$$ and $$m,k$$ variable) we have OEIS A284949 different images. So my questions are:

• How can I calculate quantity of different images for $$m,n,k$$ variable?

• How can I calculate minimum quantity of coloring choosing pixel (with smallest value of shortest way to make selected image one-colored) for any image of set, bounded by $$m,n,k$$ variable?

• How can I calculate minimum quantity of coloring any pixel (include pixel with biggest value of shortest way to make selected image one-colored) for any image of set, bounded by $$m,n,k$$ variable?