Let $ n $ be a positive natural number and let $ \mathbb N _ n = \{ 1 , \dots , n \} $. We call $ A $ a constructing set for $ \mathbb N _ n $ when $ A \subseteq \mathbb N _ n $ and for every $ a $ in $ A ^ { \mathrm c } $, there are $ b $ and $ c $ in $ A $ such that $ a = c - b $. For every positive natural number $ n $, find $ s _ n $, the minimum size of a constructing set for $ \mathbb N _ n $.
For $ n \le 29 $, I tried to find $ s _ n $ by brute force using a C program. The output was the following list of constructing sets:
$$ \begin {array} {|c|c|c|} \hline n & s _ n & \text {A Minimum-Sized Constructing Set for $ \mathbb N _ n $} \\ \hline 1 & 1 & \{ 1 \} \\ \hline 2 & 2 & \{ 1 , 2 \} \\ \hline 3 & 2 & \{ 2 , 3 \} \\ \hline 4 & 3 & \{ 2 , 3 , 4 \} \\ \hline 5 & 3 & \{ 3 , 4 , 5 \} \\ \hline 6 & 3 & \{ 2 , 5 , 6 \} \\ \hline 7 & 4 & \{ 4 , 5 , 6 , 7 \} \\ \hline 8 & 4 & \{ 3 , 6 , 7 , 8 \} \\ \hline 9 & 4 & \{ 3 , 7 , 8 , 9 \} \\ \hline 10 & 5 & \{ 4 , 7 , 8 , 9 , 10 \} \\ \hline 11 & 5 & \{ 4 , 8 , 9 , 10 , 11 \} \\ \hline 12 & 5 & \{ 4, 9 , 10 , 11 , 12 \} \\ \hline 13 & 5 & \{ 3 , 7 , 11 , 12 , 13 \} \\ \hline 14 & 6 & \{ 5 , 10 , 11 , 12 , 13 , 14 \} \\ \hline 15 & 6 & \{ 5 , 11 , 12 , 13 , 14 , 15 \} \\ \hline 16 & 6 & \{ 4 , 8 , 13 , 14 , 15 , 16 \} \\ \hline 17 & 6 & \{ 4 , 9 , 14 , 15 , 16 , 17 \} \\ \hline 18 & 7 & \{ 6 , 13 , 14 , 15 , 16 , 17 , 18 \} \\ \hline 19 & 7 & \{ 5 , 10 , 15 , 16 , 17 , 18 , 19 \} \\ \hline 20 & 7 & \{ 5 , 10 , 16 , 17 , 18 , 19 , 20 \} \\ \hline 21 & 7 & \{ 5 , 11 , 17 , 18 , 19 , 20 , 21 \} \\ \hline 22 & 7 & \{ 4 , 9 , 14 , 19 , 20 , 21 , 22 \} \\ \hline 23 & 7 & \{ 2 , 5 , 8 , 12 , 21 , 22 , 23 \} \\ \hline 24 & 8 & \{ 6 , 12 , 19 , 20 , 21 , 22 , 23 , 24 \} \\ \hline 25 & 8 & \{ 6 , 13 , 20 , 21 , 22 , 23 , 24 , 25 \} \\ \hline 26 & 8 & \{ 5 , 11 , 16 , 22 , 23 , 24 , 25 , 26 \} \\ \hline 27 & 8 & \{ 5 , 11 , 17 , 23 , 24 , 25 , 26 , 27 \} \\ \hline 28 & 8 & \{ 2 , 4 , 6 , 7 , 18 , 19 , 27 , 28 \} \\ \hline 29 & 8 & \{ 2 , 5 , 8 , 11 , 15 , 27 , 28 , 29 \} \\ \hline \end {array} $$
Other than being increasing, I can't recognize a pattern to conjecture a formula for calculating $ s _ n $. Even I couldn't prove that $ ( s _ n ) _ { n = 1 } ^ \infty $ is increasing, so it may be wrong. Furthermore, the algorithm I used is not efficient and I couldn't come up with another algorithm that is not exponential-time.
My question is whether there is a formula for $ s _ n $ and an efficient systematic way to find a minimum-size constructing set for $ \mathbb N _ n $.