I look for a reference for the following problem.
Given an integer $k$, find a set $A\subset\mathbb{N}$ with $|A|=k$
that maximizes $t$ such that $\left[0,1,..,t\right]\subset A+A$.



Sets A such that A+A contains the largest set [0,1,..,t]
A table of values for these $t$ are given in the introduction Graham and Sloane's [On Additive Bases and Harmonius Graphs](https://epubs.siam.org/doi/pdf/10.1137/0601045) (your sequence corresponds to $n\_\beta(k)$ in their notation). Graham and Sloane also give some references to previous work with this sequence, both under the name of "interval basis" (or Abschnittsbasis), going back to a paper in German from [Rohrbach](https://mathscinet.ams.org/mathscinet/search/publdoc.html?pg1=INDI&s1=149865&sort=Newest&vfpref=html&r=27&mx-pid=1545658) in the 1930's, and under the name of "The Postage Stamp Problem". 


This is sequence A001212 [in the OEIS](https://oeis.org/A001212), which has additional references. 


