# Exploring the Expanded Collatz Sequence: Is it Unique for Each Value of k?

• I
• elcaro
In summary, the conversation discusses an expanded form of the Collatz sequence, which has been explored on Wikipedia and by mathematician Erdös. The expanded sequence replaces 3n+1 with 3n+2k+1 for different values of k. The conversation raises questions about the cycles of this sequence and its uniqueness for each k value. It is suggested to watch a video on Youtube for more information on the cycle(s) of Collatz.
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?

fresh_42 said:
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.

elcaro said:
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?

## 1. What is the Expanded Collatz Sequence?

The Expanded Collatz Sequence is a mathematical sequence that starts with a positive integer and follows a specific set of rules to generate a sequence of numbers. The rules are as follows: if the number is even, divide it by 2. If the number is odd, multiply it by 3 and add 1. This process is repeated until the number reaches 1.

## 2. What is the significance of the Expanded Collatz Sequence?

The Expanded Collatz Sequence has been a subject of interest for mathematicians for decades. It is significant because it is a simple mathematical concept that has yet to be proven or disproven. It has also been used in various fields such as computer science and cryptography.

## 3. How is the Expanded Collatz Sequence related to the Collatz Conjecture?

The Expanded Collatz Sequence is an extension of the Collatz Conjecture. It is a variation of the original conjecture that includes additional rules for generating the sequence. The Expanded Collatz Sequence is often used as a tool for exploring and understanding the behavior of the Collatz Conjecture.

## 4. What is the current state of the Collatz Conjecture?

The Collatz Conjecture is still an unsolved problem in mathematics. Despite numerous attempts by mathematicians, a proof for the conjecture has not been found. However, the Expanded Collatz Sequence has provided some insights and patterns that may eventually lead to a proof.

## 5. How is the Expanded Collatz Sequence used in computer science?

The Expanded Collatz Sequence has been used in various computer science applications, such as generating random numbers and testing algorithms. It has also been used in the development of new encryption methods and in the study of chaotic systems. Its simple rules and unpredictable nature make it a useful tool for various computational tasks.

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