3n-1: More than a mirror formula of 3n+1?

[StaceyStarwolf]

Oh no, this is so complicated for me as a non mathematician, but I will try and find out what the formulas mean so I can better understand what you are saying.

I wanted to show how the 3n+1 and 3n-1 formulas intertwine and work together and got inspired by Failix his suggestion. It is based off my other artwork that shows this, but on that one I had taken the links apart so it doesn’t show that well how the formulas interact. This one does, this is the 17 loop of the x3-1. Can you spot her?

You can fill in any number and even the even ones. Both formulas are linked to all numbers, so what we get are intertwined fragments of both formulas for each number. I decided not to care about the fragments and how they interact, instead I focused on their start (x2+1 or x2-1) and end points (x3+2 or x3-2) and see where the two will line up.

For those who wonder what the other loops are: -5 is the -17 loop, -3 the -7 loop, -1 the -1 loop and of course 1 is the 1 loop, 3 the 7 loop and 5 the 17 loop. Go lower or higher and the results will expand and never link up again due to the numbers being bound by their x2 and x3 grids.

Feel free to play with this or use it as art, my art is always for free for everyone.

Edit: Funny, I can now also show that when using the 3n+1 formula rules we are ignoring the 1n+1 formula rules and compensating her results with 3n-1. And of course also vice versa:

[StaceyStarwolf]

Hey, it is me again. I am chaotic and forget a lot or accidentally delete stuff. Because I sometimes don’t know how my own confused mind works I decided to find out why I created this art piece in the Collatz art topic:

I didn’t even know what the green numbers meant. But now I think I do. Maybe you guys already figured it out, but the green numbers that I have put on the linear scale is the difference between 1n+1 and 3n+1, which is also the difference between the 1n-1 and 3n-1.

Oh no, don’t you dare!

But you already posted this Stacey, what is the difference? Yes!

See, there is a lot going on here when we are dividing. We are not only compensating our 3n+1 mistake with just 3n-1, but also with the mistake made by using 3n+1 to compensate for 3n-1!

So to avoid using the 3n formulas I apparently substituted them with the difference between the 1n and 3n formulas and as you can see there are these 2 predictable ways to do this. This makes any number either a 3n+1/4 number and thus also a 3n-1/2 number or a 3n-1/4 and 3n+1/2 number. There are no other options.

Higher numbers will have more differences and stacking results on top of each other doesn’t matter. Ups will become other ups and eventually downs due to the other formula using the same numbers to do the other thing. This will make any number always fall back into their base difference.

So looking at the base differences by putting them on a linear scale was actually a smart move. A loop will occur when two opposite numbers accidentally share the same differences when using one or both ways to avoid the 3n formulas.

For example, this is the 17 loop. Notice how both numbers have a difference of 5:

In the art piece I posted in the Collatz Art topic and here you can find the other loops and how they work and that the results fan out making sure there are no other loops besides the one we now know. I think it is because the differences become to big to be able to avoid the x3 part of the 3n formulas.

I am sorry for being chaotic again. I just wanted to elaborate on something I created but don’t really understand, because of all the things that are going on in my mind. But hopefully my art explains it and I hope you enjoyed my silliness again!

[StaceyStarwolf]

It has been awhile since I revisited this topic and I think this is the best place to post some of my new shenanigans.

I was playing with the he T (n) = 3n+1/2 (n odd) and T (n) = 3n-1/2 (n odd) formulas and found out that they work in pairs. For example:

For every other number they are helping each other reach their outcomes and they also mirror previous formula pairs.

However, the interesting thing is that there are whole new Collatz Conjectures behind the ones we are using. Instead of blindly dividing n by 2 we could follow the paired formula.

That would mean instead of T (n) = 3n+1/2 (n odd) and n/2 (n even) this becomes T (n) = 3n+1/2 (n odd) and n*-1/2 (n even) and for T (n) = 5n+1/2 (n odd) and n/2 (n even) this becomes T (n) = 5n+1/2 (n odd) and n*-3/2 (n even) and so on.

They are pretty interesting and nicely balanced Conjectures that I think proper mathematicians could look into, especially if we can prove they are responsible for connecting all the numbers for the ones we are researching now.