A graphical representation of the final digit resulting from iterations of the product of the two numbers formed from the alternating digits of n

Question

Sorry for the extremely long title. I did not know exactly what to put in the title since the question I want to pose is rather vague (and so I shall also accept vague answers, hehe...). The purpose of this thread is to see whether someone wants to "follow along" - so to speak - on my little experiment and see if we can generate something interesting. So please, bare with me, as I build up to my question.

I've been doing some recreational math recently and started playing around with the sum of the product of the digits in a number $n$. Or, as OEIS formulates it:

"Final digit resulting from iterations of the product of the two numbers formed from the alternating digits of n" (see A087471).

Actually I dont know if the formulation by OEIS is equivalent to what I've been doing but when I put in the sequence generated by "my algorithm" into OEIS it outputs the sequence A087471. Anyhow...

The algorithm goes as follows: You pick a number, say $13556$, then add the product of the digits in the following sense $1\cdot 3+3\cdot 5+5\cdot 5+5\cdot 6=73$ and you do the same thing with $73$ and get $7\cdot 3=21$ and lastly with $21$ and get the one-digit number $2\cdot 1=2$.

So I used Mathematica to generate a list of these final digits for all positive whole numbers $i$ in the interval $10\le i\le n$, where $n$ is (usually) some big number. After this I wanted to somehow graphically investigate these final digits so I created a color coded array where each digit $0,1,2,3,4,5,6,7,8,9$ got a different color as shown in the figure below which shows that $10$ results in $0$, $11$ results in $1$, etc...

Now when creating this array for larger values of $n$ (say $n=2000$) one gets a pretty striking (in my view) image which is shown in the figure below.

As one can see there is some kind of structure for smaller values of $n$ up to and including $n=1200$ which to me looks something like a "bullseye" moving to the left, hehe. But one also observes that this "structure" starts to dissipate for larger values of $n$. I wonder: why this is the case? Is there some concrete way of describing the behaviour of this algorithm which can elucidate this visual pattern?

Thanks in advance.