On $s$-additive sequences

Question

For a non-negative integer $s$, a strictly increasing sequence of positive integers $\{a_n\}$ is called $s$-additive if for $n>2s$, $a_n$ is the least integer exceeding $a_{n-1}$ which has precisely $s$ representations $$a_i+a_j=a_n$$ for $1\le i<j<n$.

The first $2s$ terms are called the base of the sequence.

A sequence is called regular if successive differences $a_{n+1}-a_n$ are eventually periodic. In other words, there is a positive integer $N$ such that $$a_{N+n+1}-a_{N+n}=a_{n+1}-a_n$$ for sufficiently large $n$. The smallest such $N$ is called the period of the sequence.

I found the article Are 0-Additive Sequences Always Regular? which talks about a specific case of an $s$-additive sequence. It mentions two results by Raymond Queneau, namely that \begin{align*} \mathcal P_0(1,k) = \begin{cases} 1 &\text{if $k$ is odd or $k=2$}\\ 4 &\text{if $k=4$}\\ 3 &\text{if $k=6$}\\ k+3 &\text{if $k\ge 8$ is even} \end{cases} \end{align*} and \begin{align*} \mathcal P_0(2,k) = \begin{cases} 2 &\text{if $k=3$ or $k=8$}\\ 1 &\text{if $k=4$}\\ k &\text{if $5\le k\equiv 1 \pmod 4$}\\ k+1 &\text{if $6\le k\equiv 2 \pmod 4$}\\ \frac{3k+3}4 &\text{if $7\le k\equiv 3 \pmod 4$}\\ \frac{3k}4 &\text{if $12\le k\equiv 0 \pmod 4$} \end{cases} \end{align*} where $\mathcal P_0(a,b)$ is the period of the $0$-additive sequence with base $\{a,b\}$.

Finch also remarks that from "the sheer magnitude of" Queneau's computations, it can be conjectured that all $0$-additive sequences are regular.

Now, the given reference is Sur les Suites $s$-additives which is unfortunately written in French (which I do not understand).So, one of the three main questions I want to ask in this post is whether there are any English translations of this paper available. If not (which seems to be more likely), then are there any expository articles written about it (in English) from which I can see the methods that Queneau used to establish this?

Apart from that, how much progress have been made on these questions so far? Also, have people considered the analogous problem of starting with $\{a_1=x,a_2=y\}$ and at the $n$-th step appending the smallest integer greater than $a_{n-1}$ which cannot be written as a sum of the $a_k$'s? If yes, then what are some of the main references for this?

---

Some more references :-

1. OEIS
2. Conjectures about $s$-additive sequences
3. Patterns in $1$-additive sequences
4. On the regularity of certain $1$-additive sequences
5. A class of $1$-additive sequences and quadratic recurrences
6. The unreasonable rigidity of Ulam sequences
7. Rigidity of Ulam sets and sequences

---

Also posted on MathOverflow (now bountied)