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.

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.

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:

• $52 \to 26 \to 13$

• $10 \to 5 \to 16$

• $13 \to 40$

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.

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 $x_0 = 0$.