---
description: Looking the other way
---

# Inverse Canonical Form

ECFs provide a clean form to analyze trajectories, so much so that the entire trajectory can be represented by a single positive integer. However, they do not provide much algebraic power we require.&#x20;

**Inverse Canonical Form** (ICF) is yet another form to represent trajectories that provide great algebraic properties. Again, consider a sequence:

$$
n \xrightarrow{p_0} R_0(n) \xrightarrow{p_1 - p_0} R(n_0) \xrightarrow{p_2 - p_1} R(n_1) \xrightarrow{p_3 - p_2} \ldots \xrightarrow{p_m - p_{m-1}} R(n_{m-1})
$$

with ECF $$\{p_0, p_1, p_2\ldots, p_m\}$$. Denote the final number in this trajectory as $$n_m = R(n_{m-1})$$, we will also refer to this number as the **terminating number**. The ICF for this trajectory is an equation, shown as:

$$
n = n_m\frac{2^{p_{m}}}{3^{m}} - \frac{2^{p_{m-1}}}{3^{m}} - \frac{2^{p_{m-2}}}{3^{m-1}} - \ldots - \frac{2^{p_1}}{3^2} - \frac{2^{p_0}}{3^1}
$$

Unlike ECF, ICF requires $$n$$ to be defined for the form. ECF only required the trajectory itself, not the starting number; ICF needs it for the left hand-side of the equation. $$n_m$$ can be computed from the trajectory. Same logic applies to giving $$n_m$$ and computing $$n$$ in reverse, but for convenience we require $$n$$ to be given.

Note that the leftmost two terms have the same denominator $$3^m$$, that is not a typo!

{% hint style="info" %}
Sequence of 3 has ECF $$\{0, 1, 5\}$$, then the ICF is:

$$
3 = 1\times\frac{2^5}{3^2} - \frac{2^1}{3^2} - \frac{2^0}{3^1}
$$
{% endhint %}

ICF has been discovered before too, with slightly different notation, see [https://www.uni-kassel.de/fg\_pur/helms/math/collatz/aboutloop/CollatzLoopIntro\_General.htm.](https://www.uni-kassel.de/fg_pur/helms/math/collatz/aboutloop/CollatzLoopIntro_General.htm)

