This is my first time posting on here so apologies if I'm breaking formatting rules. Also, I'm not a native English speaker, yadda yadda.
So a few years ago I found something in school when the lesson we usually had was cancelled. I was playing around with my calculator, found that it had a random integer generator and wrote down a list of random numbers between 1 and 6.
Underneath that list I counted how many times the number above had already appeared before that modulo 6, and I wrote that down in the second row. (I was really bored). I repeated the process a couple of times and noticed something that I found really strange and surprising: after one iteration, the rows were identical alternatingly, if you know what I mean. So the second row was identical to the fourth, the third to the fifth, the fourth to the sixth and so on.
Now, a few years later and I'm in the third year of high school. My maths teacher now is an absolutely brilliant man, and a great teacher. I wish I had him earlierm he's the kind of person who can really get people excited about maths. Anyways, I was thinking about this a few weeks ago and I thought: "Why don't I ask him, he'll surely be able to at least help me understand why this happens". And so I did.
To make a long story short: I'm trying to notate this rigourously. Here's what I came up with (it's most likely wrong, but bear with me here)
$A_{n+1}(x) = \sum_{y=1}^{x}A_{n}(y) = A_{n}(x) \mod \max(A_{0}) $
So that is indeed quite a mess. Here's what I mean:
"In the list $\ A_{n+1}$, the x-th element is defined as the sum of every time that every element before the x-th one is equal to the y-th, y being a number that counts up to x. That all modulo the biggest number that was in the original list, which is$\ A_{0}$."
I think an example would be useful here.
Say:
$\ A_{0} = [2, 4, 6, 5, 5, 2, 4, 3, 1, 1, 5, 6, 3] $
That would make:
$\ A_{1} = [1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 3, 2, 2] $
And as a result,
$\ A_{2} = [1, 2, 3, 4, 1, 2, 3, 5, 6, 4, 1, 5, 6] $
And the claim is thus that $\ \forall x \geq 1 : A_{x} = A_{x+2} $
So, that's my question. How do I notate this?
I hope I wasn't being confusing. Thanks in advance.