Minimum number of elements in ${0, 1, 2, dots, n}$ that add up to all of the elements of ${0, 1, 2, dots, n}$.

Maybe you should use a different approach.

It can be found that sets with the least number of elements are of the following type:

${0,1,a_1,dots ,a_m,frac{n}{2}-a_m,dots ,frac{n}{2}-a_1,frac{n}{2}-1,frac{n}{2}}$

or

${0,1,a_1,dots ,a_m,frac{n}{4},frac{n}{2}-a_m,dots ,frac{n}{2}-a_1,frac{n}{2}-1,frac{n}{2}}$

example $n leq 20$

elements can be found easily with this set

${0,1,3,dots ,frac{n}{2}-1,frac{n}{2}}$ number of elements $2+frac{n}{4}$

then

${0,1}$ for $0 leq n leq 2$

${0,1,2}$ for $3 leq n leq 4$

${0,1,3,4}$ for $5 leq n leq 8$

${0,1,3,5,6}$ for $9 leq n leq 12$

${0,1,3,5,7,8}$ for $13 leq n leq 16$

${0,1,3,5,7,9,10}$ for $17 leq n leq 20$

example $n >20$

${0,1,3,4,9,10,12,13}$ for $21 leq n leq 26$

${0,1,3,4,5,8,14,20,26,32,35,36,37,39,40}$ for $73 leq n leq 80$

You can solve the problem via integer linear programming as follows. For $jin S$, let binary decision variable $x_j$ indicate whether element $j$ is selected. Let $P={j_1in S, j_2 in S: j_1 le j_2}$ be the set of pairs of elements of $S$. For $(j_1,j_2)in P$, let binary decision variable $y_{j_1,j_2}$ indicate whether both $j_1$ and $j_2$ are selected. The problem is to minimize $sum_{jin S} x_j$ subject to: begin{align} sum_{(j_1,j_2)in P:\j_1+j_2=i} y_{j_1,j_2} &ge 1 &&text{for $iin S$} tag1\ y_{j_1,j_2} &le x_{j_1} &&text{for $(j_1,j_2)in P$} tag2\ y_{j_1,j_2} &le x_{j_2} &&text{for $(j_1,j_2)in P$} tag3 end{align} The objective minimizes the number of selected elements. Constraint $(1)$ forces each element of $S$ to be expressible as a sum of selected elements. Constraints $(2)$ and $(3)$ enforce $y_{j_1,j_2} = 1 implies x_{j_1} = 1$ and $y_{j_1,j_2} = 1 implies x_{j_2} = 1$, respectively.