If the initial state of Conway's game of life is a line of $n \in [0,100]$ alive cells, then it vanishes completely after some steps iff $n \in \{0,1,2,6,14,15,18,19,23,24 \}$. See below for $n=24$.

*Question*: Is such a line non-vanishing for any $n \in [25,\infty)$?

**Edit**

*Definition*: a finite pattern $p$ has a *weak period* $wp$ if for any cell $c$ in the grid, there is $k>0$ such that the set of cells which are neighbours of neighbours of neighbours... ($k$ times) of $c$, is periodic of period $wp$ after sufficiently many generations, from the initial state $p$.

The sequence A061342 gives the weak period $wp_n$ of a line of $n$ alive cells. By combining the checking above with the fact that $wp_n \ge 2$ for $n \in [84,1000]$, we deduce that the pattern is non-vanishing for $n \in [25,1000]$. We observe that for $n=500$, four gliders are produced on the boundaries after $435$ steps, but $435<500$, so this must happens $\forall n \ge 500$. *Assuming* that these gliders (or others) are perpetual (as stated *implicitly* by Nathaniel Johnston in A061342, although without reference, while the proof could be non-trivial, as pointed out by Will Sawin in the comments), the answer to the above question would be **yes**.

*Definition*: a finite pattern $p$ is *weakly-vanishing* if any cell $c$ in the grid becomes perpetually dead after sufficiently many generations (depending on $c$), from the initial state $p$.

**Improved question**: Is there a weakly-vanishing line of $n$ alive cells with $n \in [25,\infty)$?

*Stronger question*: Is $wp_n \ge 2$ for any $[84,\infty)$ ?

Tobias Fritz pointed out in the comments that there is a one-cell thick pattern with infinite growth (see this page), but it is disconnected. *Bonus question*: Can that happen in the connected case?

stationary component? Do you know that the pattern can be decomposed for long times into gliders/spaceships plus something that happens in a bounded region? This is not always the case, e.g. there even are patterns that grow quadratically (pentadecathlon.com/lifeNews/2011/05/…). $\endgroup$ – Tobias Fritz Dec 13 '17 at 18:05