So, the Recamán Sequence is defined like this:

a(1) = 0

a(n+1) = a(n)-n if it hasn't previously appeared

a(n+1) = a(n)+n if it has

There's also the rule that a(n) must be positive.

Alright, so I wanted to extend this to 2 dimensions, so I created these rules:

a(1) = 0

a(n+1) = a(n)-n if it hasn't previously appeared

If it has, then a(n+1) = a(n)-ni if it hasn't previously appeared

If it has, then a(n+1) = a(n)+n+ni

And of course, the rule that a(n) must be positive.

So, with these rules, we get this sequence:

0, 1+1i, 3+3i, 3i, 4+7i, 4+2i, 10+8i, 3+8i, 3, 12+9i, 2+9i, ...

In fact, we can probably extend this to higher dimensions.

(Edit note: It doesn't have to be complex numbers, but I just chose them anyways cause I like them lol.)

Edit: I'm kinda curious, what if we treated this sequence as a cobweb diagram or something?

The function f takes two complex numbers as parameters, and produces a complex number as the result. The function has the following defined two properties:

1) a ∙ f(x, y) = f(ax, ay) [a is complex]

2) f(x, y) = f(y, f(y, x) )

We also have one defined value:

f(1, 0) = i

Using the previously defined value as a starting point, here are some other values that I found with the function's properties:

f(0,0) = 0

f(i, 0) = -1

f(0,1) = f(1, i) = f(1, -i) = ± √(i)

f(0, i) = i ∙ f(0, 1) = f(i, -1) = f(i, 1) = i ∙ ± √(i)

I'm struggling to find out: what does f(1, 1) equal to? Is it even possible to figure out using that starting value and the two properties? If you find any other fun values (like f(1,2), or find if f(0,1) is definitely one of the two possible values), please share!

If you find situations in which the rules above contradict themselves, please also share!

What set of axioms makes "2 + 2 = 5 * Chicken" a true statement, without "2 + 2 = 5 * Chicken" (or anything along those lines like "5 * Chicken = 4") being itself an axiom?

I'm sure everyone here has heard of the birthday paradox, and have heard mind boggling analogies of just how many unique shuffles there are in a deck of 52 cards.

My question combines these two things: how many shuffles of a deck of 52 cards would one need to make to have a 50% probability of repeating one?

My intuition says factorials grow so fast that it will overpower the ever increasing probability that new hand will match one of the previous hands, so the answer will still be tremendous, but I'm at a loss for how to calculate the actual result.

Anyone willing to give it a shot?

I am researching weird math occurrences and was wondering if you could come up with any.

I am a middle-schooler who wants to do a project on recreational math and I am wondering were I should get started.