Integer sequences with unique $k$-subsets sum

Question

let the $\binom{\mathfrak{M}}{k}$ be a shorthand notation for chosing $k$ elements of set $\mathfrak{M}$ of positive integers and let $\left|\binom{\mathfrak{M}}{k}\right|$ denote the sum of the selected number's values and depending also the number with that sum as its value; consequently $\Big\lbrace\left|\binom{\mathfrak{M}}{n}\right|\Big\rbrace$ is the set of all value sums of $k$ elements of $\mathfrak{M}$

Questions:

defining a sequence $\mathfrak{M}_i$ of sets of positive integers via:
\begin{align}\mathfrak{M}_{0\phantom{1+i}} &:= \lbrace 1,\,\dots,\,k\rbrace \\ \mathfrak{M}_{i+1}\ &:=\ \mathfrak{M}_i\ \cup\ \min\limits_{n\in\mathbb{N}}:\,\ \Bigg\lbrace\left|\binom{\mathfrak{M}_i}{k}\right|\Bigg\rbrace\bigcap\Bigg\lbrace\left|\binom{\mathfrak{M}_i}{k-1}\right|+n\Bigg\rbrace=\emptyset\end{align}

• what is the complexity of determining the sequence of values $n$ whose union with $\mathfrak{M}_i$ yields $\mathfrak{M}_{i+1}$

• what is the growth rate of the sequence of maximal elements $n_i$ of $\mathfrak{M}_i$

• for $k=2$ is it true that $\mathfrak{M}_i$ is the set of the first $i$ non-equal Fibonacci numbers?

Answer 1

After having calculated the sequence for $k=2$ with this python program

from itertools import combinations

sequence = set([1,2,3])
pairs ={sum(list(x)) for x in combinations(sequence,2)}

for i in range(4,100):
    candidates = {x+i for x in sequence}    
    if candidates&pairs != set():
        ++i
    else:
        sequence |= {i}
        pairs    |= candidates

print(sorted(sequence))

and entering the generated output

[1, 2, 3, 5, 8, 13, 21, 30, 39, 53, 74, 95]

into the OEIS, it turned out that it is already known as sequence A011185 and is not the sequence of Fibonacci numbers with $30$ being the first non-Fibonacci number.