Now, when I talk about a tricot as I did several times here, I am talking about one of those affine Collatz-like problems. A tricot is defined by a matrix M of width 2, in that case h is the height of that matrix and for any i \in \{0, 1 ... h-1\}, f_i(x)=M_{0,i}x+M_{1,i}
From a tricot T=tricot\begin{pmatrix}
N_0 & a_0 \\
\vdots & \vdots \\
N_i & a_i \\
\vdots & \vdots \\
N_{h-1} & a_{h-1}
\end{pmatrix}, we can extract the characteristic application:
App_T(x) = f_{map_T(x)}(reduce_T(x)) = f_{x\%h}(\lfloor x/h \rfloor) = N_{x\%h}\lfloor x/h \rfloor+a_{x\%h}
And thus, define the flight of x in T as:
flight(x,T)=[App_T^i(x)]_i
The trajectory can be deduced from it:
traj(x,T) = [map_T(App_T^i(x))]_i = [App_T^i(x) \% h]_i
For example, standard-Collatz is represented by tricot\begin{pmatrix} 1 & 0 \\ 6 & 4 \end{pmatrix}.
Reduced-Collatz is represented by tricot\begin{pmatrix} 1 & 0 \\ 3 & 2 \end{pmatrix}.
“5n+1” is represented by tricot\begin{pmatrix} 1 & 0 \\ 10 & 6 \end{pmatrix} for its standard version and tricot\begin{pmatrix} 1 & 0 \\ 5 & 3 \end{pmatrix} for the reduced one.