It was proven by Gauss that every integer is the sum of at most three triangle numbers, and by Lagrange that every integer is the sum of at most four squares.
The triangles are the set $\big\{1,3,6,10,\ldots\big\}$, and it's easy to see that the squares can be written as $\big\{1,1\!+\!3,3\!+\!6,6\!+\!10,\ldots\big\}$.
I wonder if it's generally the case that if a set $S$ (with $s_n$ the $n^{\mathrm{th}}$ element) is a $p$-additive base for the integers, then the set $S^*=\big\{s_1,s_1+s_2,s_2+s_3,\ldots\big\}$ is (at most?) $p+1$ additive.
