I have a straightforward question about Conway's "Look and Say Sequence (A005150):
The integer sequence beginning with a single digit in which the next term is obtained by describing the previous term. Starting with 1, the sequence would be defined by "1, one 1, two 1s, one 2 one 1," etc., and the result is 1, 11, 21, 1211, 111221, ....
Source: http://mathworld.wolfram.com/LookandSaySequence.html
Question: Is this sequence strictly increasing?
It is known that, asymptotically, the number of digits in each term grows by a little more than 30%:
if $L_n$ is the number of digits in the $n$-th term in the sequence, then
$\lambda := \displaystyle{\lim_{n\to\infty}\frac{L_{n+1}}{L_n} = 1.303577269\dots}$
where $\lambda$ is the unique positive real root of the following degree-71 polynomial: [...]
Source: http://www.nathanieljohnston.com/2010/10/a-derivation-of-conways-degree-71-look-and-say-polynomial/
However, is this growth monotonic everywhere in the sequence, or do there exist a finite number of pathological examples where the $(n+1)$-th term is smaller than the $n$-th term?
Surely, this depends on the initial seed of the sequence. For example, under the "look and say" operation, $111222\to 3132$ decreases. Is it known whether there are any instances of behavior like this in the sequence where the seed is 1?