TITLE: Numbers of distinct values obtained by inserting $+ - \times \div ()$ in $\underbrace{2\quad2 \quad2 \quad2\quad...\quad 2}_{n \text{ times}}$
QUESTION [25 upvotes]: This question is inspired by How many values of $2^{2^{2^{.^{.^{.^{2}}}}}}$ depending on parenthesis? (By the way, I sincerely hope this kind of questions can receive more attention) 

Insert $+ - \times \div ()$ in 
  $$\underbrace{2\quad2 \quad2 \quad2\quad...\quad 2}_{n \text{ times}}$$
  Denote the number of distinct values which can be obtained in this way by $D(n)$.
  Is there a general formula (or recurrence relation at least) for $D(n)$?

This is basically the $+ - \times \div ()$ version of @barakmanos question. It seems this question is easier than the power tower version. Or maybe not? 
For $n=1$ , there is only $2$ values $-2,2$;
For $n=2$, there are $5$ values $-4,-1,0,1,4$;
For $n=3$, there are $13$ values $-8,-6,-3,-2,-1,-\frac{1}{2},0,\frac{1}{2},1,2,3,6,8$;
And for $n=4$ I'm reluctant to calculate with bare hands. (See @DanUznanki answer for what follows)
Any idea is appreciated. Sorry if this is a duplicate.
Edit: My research shows that the version with distinct generic variables $a_1,a_2,...,a_n$ is solved. See A182173 for your reference.

REPLY [3 votes]: @Dan Uznanski if your results are correct i noticed that the ratio between the successive terms of the sum is almost constant. So maybe:
Conjecture
$$D(n) \sim K^n $$
Where:
$$2.3\leq K \leq 2.5 $$