Consider the following sequence (A050000 in the OEIS): $$k(n)=\left\{\begin{array}{ll} \lfloor{0.5\cdot k(n-1)}\rfloor, & \lfloor{0.5\cdot k(n-1)}\rfloor\notin \{0, k(1), k(2), \dots, k(n-1)\} \\ 3\cdot k(n-1) & otherwise\end{array}\right.$$ with $k(1)=1$. Does it ever repeat, i.e. are there two distinct natural numbers $i, j$ such that $k(i)=k(j)$? And does the sequence contain every positive natural number?
As for the first part, I already tried assuming there were two such natural numbers and then picked the pair $(i, j)$ with the smallest $j$ and tried to reduce this to a contradiction, e.g. there is a pair $(i', j')$ with $k(i')=k(j')$ with $j'<j$. This works fine except for the case that $k(i)= \lfloor{0.5\cdot k(i-1)}\rfloor$ and $k(j)=3\cdot k(j-1)$. That's where I'm stuck at the moment.
Can anyone give a $\textbf{hint}$?
EDIT: Concerning the second question: If $K$ is the set of all the elements of the sequence and we suppose that there is some natural number $M$ with $M \notin K$, then let $m$ be the smallest such $M$. It follows that $2m$ and $2m+1$ cannot be in $K$, otherwise they would generate $m$ as the next element. Likewise, $4m, 4m+1, 4m+2, 4m+3 \notin K$ and $8m, 8m+1, 8m+2, ..., 8m+7 \notin K$. In general $2^lm, 2^lm+1, 2^lm+2, ..., 2^lm+(2^l-1) \notin K$ for every nonnegative integer $l$. Hence, there are arbitrarily long "gaps" in $K$ if there was one natural number that is not in $K$. Does this help to answer the second question? Does it maybe help if we suppose that we have already proved that the sequence does never repeat (first question)?