Let $A=\{a_1,a_2,\dots,a_n\,\vert\,a_1\lt a_2\lt\cdots\lt a_n\}$ be a finite subset of $\Bbb N$ with sumset $$A+A=\{a_i+a_j\,\vert\, a_i,a_j\in A\}$$ What is the longest possible chain of consecutive numbers in $A+A$ ? I know that if $|A|=n$ then $$2n-1\le |A+A|\le\frac{n(n+1)}{2}$$ but are there better bounds or an exact formula? The lower bound can be improved to $\frac{(3+n)^2-8}{8}$ by placing the numbers in a configuration more or less efficient (but I think not optimal)
The first few values for $n=1,2,\dots,7$ are $1,3,5,9,13,17,21$