Consider the strictly increasing integer sequence $f(n)$ defined as :
$$f(1) = 1$$
$ M $ is in the list If and only if $M$ is not of the form $ f(a) + f(b) + f(c) $ for Some integers $a,b,c > 0$.
So the sequence starts as
$$1,2,7,8,...$$
This resembles the A-sequence.
http://mathworld.wolfram.com/A-Sequence.html
What is known about $f(n) $ ? In particular : How fast does $f(n)$ grow ?
The sequence continues $$1,2,7,8,13,14,19,20,...,6n+1,6n+2,...$$
The integers in the sequence are all of the form either $6n+1$ or $6n+2$. When these are added together in threes the sums that can be made are restricted to the forms $6n+3,6n+4,6n+5$ and $6n$. Since all such positive integers can be made, none of these appear in the sequence. Conversely, integers of the form $6n+1$ and $6n+2$ can never be the result of a sum.
Hence the sequence contains all positive integers of the form $6n+1$ and $6n+2$ and no others. Therefore, the function grows linearly.