Inverse Canonical Form

Looking the other way

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.

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!

Sequence of 3 has ECF $\{0, 1, 5\}$, then the ICF is:

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

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.