P8742 [Lanqiao Cup 2021 NOI Qualifier AB] Weights on a Balance

You have a balance scale and $N$ weights. The weights have masses $W_{1}, W_{2}, \cdots, W_{N}$ in order. Please compute how many different weights can be measured in total. Note that the weights can be placed on both sides of the balance.

The first line contains an integer $N$. The second line contains $N$ integers: $W_{1}, W_{2}, W_{3}, \cdots, W_{N}$.

Output one integer representing the answer.

**[Sample Explanation]** The 10 measurable weights are: $1 、 2 、 3 、 4 、 5 、 6 、 7 、 9 、 10 、 11$. $$ \begin{aligned} &1=1 \\ &2=6-4(\text { place } 6 \text { on one side of the balance, and } 4 \text { on the other side) } \\ &3=4-1 \\ &4=4 \\ &5=6-1 \\ &6=6 \\ &7=1+6 \\ &9=4+6-1 \\ &10=4+6 \\ &11=1+4+6 \end{aligned} $$ **[Scale and Constraints for Test Cases]** For $50 \%$ of the test cases, $1 \leq N \leq 15$. For all test cases, $1 \leq N \leq 100$, and the total weight of the $N$ weights does not exceed $10^5$. Lanqiao Cup 2021 First Round Provincial Contest, Group A Problem F (Group B Problem G). Translated by ChatGPT 5