Trace of sequence
Denote by $\mathbb{N}=\{0,1,2,...\},~$ the set of natural numbers, and by $I_{m}=\{0,1,...,m-1\}\,$ the set of natural numbers lesser than given natural number $m$. Let $c=(c_0,c_1,...,c_{m-1})\,$ a $m$-sequence of natural numbers, and $p=max\{c_0,c_1,...,c_{m-1}\},\,$ the greatest term of sequence $c$
Then the sequence
$t(c)=(t_0,t_1,...,t_p)\,$
where $t_j,j\in I_{p+1}\,$ denote number of terms of sequence $ c$ thats are equal at $j$, is called trace of $c$. It is clear that terms of trace fulfills the conditions
$t_0+t_1+...+t_p=m\,$
$t_1+2t_2+...+pt_p=c_0+c_1+...+c_{m-1}\,$
Denote by
$t^{0}(c)=c\,$
$t^{n}(c)=t(t^{n-1}(c))\,$
1.The set of sequences
$B=\{(1,0,0,1),(2,2),(0,0,2),(2,0,1),(1,1,1),(0,3)\}\,$
that is cycle of length 6 is called '''bracelet of sequences''' because for each sequence $c$ from $B$ holds
$t^6 (c)=c\,$
2.The set of sequences
$R=\{(0,1,1),(1,2)\}\,$
that is cycle of length 2 is called '''ring of sequences''' because for each sequence $c $ from $R$ holds
$t^2 (c)=c\,$
The set
$ H=B\cup R\,$ is called ''' black hole of sequences'''
Reasons for that name are because I suppose that:
Claim: For each finite sequence $a$ of natural numbers exists natural number $n$ such that $t^n(a)\in H\,$ in other words each sequence converges to $H$.
Sequence $a$ is of type $B$ if its converge to $H$ from $B$ for example sequence $(2,3)$ is of type $B$ because
$t^3((2,3))=(0,0,2)\in B\,$
And sequences that converges to $H$ from $R$ are of type $R$ for example the sequence $(0)$ is of type $R$ because
$t^6((0))=(1,2)\in R\,$
My questions are.
- Is my assumption true
- If it is true how to decide of which type is any given finite sequence of natural numbers
- Can be done any programme or algorithme to determine of which type is certain sequence. Thanks