# Expanded Collatz Sequence

• I
elcaro
TL;DR Summary
Expanding the Collatz sequence by replacing 3n+1 with 3n+2k+1 for k=0,1,2,...
For k=0 we get the original sequence, leading to the cycle 4-2-1. If the Collatz conjecture holds, that would be true for all integer values of n>0.
For values of k>0 we get different cycles.
For k=1 we for instance get the cycle 3-12-6-3
What we want to investigate is:
- What cycle is the sequence iterating to for different values of k?
- Is that cycle unique for k (independent of the seed number)?
Has this expanded Collatz sequence been explored previously?

elcaro
I haven't seen this one. But even Wikipedia lists so many generalizations
https://it.wikipedia.org/wiki/Congettura_di_Collatz
https://de.wikipedia.org/wiki/Collatz-Problem
https://en.wikipedia.org/wiki/Collatz_conjecture
that yours might have been among them. Compare especially the Syracuse function.

I am with Erdös:
I think my expanded form of the Collatz sequence is already covered as a special case of the natural generalization of the Collatz sequence, explored by Conway.

Bert-W
TL;DR Summary: Expanding the Collatz sequence by replacing 3n+1 with 3n+2k+1 for k=0,1,2,...
For k=0 we get the original sequence, leading to the cycle 4-2-1. If the Collatz conjecture holds, that would be true for all integer values of n>0.
For values of k>0 we get different cycles.
For k=1 we for instance get the cycle 3-12-6-3
What we want to investigate is:
- What cycle is the sequence iterating to for different values of k?
- Is that cycle unique for k (independent of the seed number)?

Has this expanded Collatz sequence been explored previously?