> For the complete documentation index, see [llms.txt](https://erhany96.gitbook.io/collatz-prefixes/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://erhany96.gitbook.io/collatz-prefixes/theory/trajectory-and-sequence.md).

# Trajectory & Sequence

There will be many numbers visited as we traverse from some number $$n$$ all the way down to 1. Consider these visited numbers in the order they appear, any slice of this list of numbers will be a trajectory. We have a handy notation for this as follows:

$$
n \to F^1(n) \to F^2(n) \to \ldots \to F^m(n)
$$

If the **trajectory** ends with 1, we call this to be a **sequence**. 

{% hint style="warning" %}
Note that a sequence can continue indefinitely after 1, and repeat many 1s until finally terminating. For our convenience, a **sequence** will refer to a **trajectory** that ends with 1 and does not repeat.
{% endhint %}

{% hint style="info" %}
For example, the number 17 has the following sequence:

$$
17 \to 52 \to 26 \to 13 \to 40 \to 20 \to 10 \to 5 \to 16 \to 8 \to 4 \to 2 \to 1
$$

Any *slice* of this could be considered a trajectory:

* $$52 \to 26 \to 13$$
* $$10 \to 5 \to 16$$
* $$13 \to 40$$
  {% endhint %}

## Reduced Trajectories

As we can see, trajectories can be rather lengthy; even with small starting numbers. We have already mentioned that looking at odd numbers suffice, and indeed they will make analyzing these trajectories a lot prettier too. The notation will be similar to before:

$$
n \xrightarrow{x\_0} R\_0(n) \xrightarrow{x\_1} R(n\_0) \xrightarrow{x\_2} R(n\_1) \xrightarrow{x\_3} \ldots \xrightarrow{x\_m} R(n\_{m-1})
$$

Here, the notation works as follows:

* $$n\_0 = R\_0(n)$$ and $$x\_0$$ is the power of two used within $$R\_0$$.
* $$n\_i = R(n\_{i-1})$$ and $$x\_i$$ is the power of two used within that $$R$$.

Likewise, if the last number in the **reduced trajectory** is 1 then this is called to be a **reduced sequence**.

{% hint style="info" %}
Looking again at number 17, its reduced sequence this time:

$$
17 \xrightarrow{0} 17 \xrightarrow{2} 13 \xrightarrow{3} 5 \xrightarrow{4} 1
$$

Note that 17 appears twice at the start, because it is already an odd number and $$R\_0$$ will not do anything, thus $$x\_0 = 0$$.
{% endhint %}