"transition" blending adjacent blocks of gray

I'm new to the list and not an expert on image terminology, so I
apologize if my post is confusing.

How can I take a 100x100 light gray square, overlap it by 20 pixels with
a 100x100 dark gray square, and have a smooth transition between the two
shades of gray.  I don't want the 20x100 pixel transition to be medium
gray, I want it to smoothly transistion from light to dark gray.

My ultimate goal:

I'm trying to represent a matrix of numbers as an image.  The numbers
will be from 0 to 1, and each number will be represented as a 100x100
square in a shade of gray.  The idea is that a person can look at the
image and get an impression of the values in the matrix by how light or
dark the image is.  I think hard transitions between squares will be
distracting, so I'd like smooth transitions.

Thanks for any advice you can give.

-Rob