After browsing the ever excellent Don Steward’s Median site for sequences, I found this on Linear and Quadratic growths.

It got me thinking about using shapes to represent sequences and in particular using different colours to represent different sequences laid on top of each other. For example, the following patterns produce a linear sequence.

However, there are a couple of ways you could look at this which combine sequences which are arguably simpler.

Or:

We can then extend this on to quadratic sequences. This is nice: one way we can see the sequence of square numbers, the other way we can see how multiplying one dimension by the other leading to some brackets which can be multiplied out.

There are a bunch of these in this file.

There are also over a hundred sequences at visualpatterns.org a website entirely dedicated to, well, visual patterns.

Get your students to create their own once they have got the idea. Could make for some great wall displays!

On quadratic sequences specifically, a nice worksheet here from @solvemymaths.

Finally, back to Don Steward with this which is actually just a series of terms stacked on top of each other.