Convex figures on triangular grid

I try to solve this question :
How many convex figures of given area are there ?
The figures should be polygons formed by these types of lines :

The possible figures formed by the first (2nd) type of lines I denote A_n (B_n).
n - is the area of figure, unit (U) is triangle on the picture above.
This choice of possible lines is equivalent to the choice

Another possibility is to study convex polydrafters (D_n)

Relations

B_4n < A_n < B_n
A_n < D_n < A_4n

Images

A_2      A_4      

A_6     A_8    A_8*

A_12   A_16    A_20

B_6      B_8      B_12 

D_2    D_3     D_4

D_5     D_6

A_8

A_12

B_16

D_12

X* is set X without the longest piece

some other images

n A_n B_n D_n A_4n B_4n B_16n  
1 0 0 1 3 0 6  
2 2 0 4 8 3 14  
3 0 0 3 11 1 23  
4 3 0 7 17 6    
5 0 0 7 21 0    
6 5 1 13 31 9    
7 0 0 9 33 1    
8 8 3 15 45 14    
9 0 0 9 52 3    
10 7 0 14 63 12    
11 0 0 12 72 1    
12 11 1 27 91 23    
13 0 0 19        
14 11 0 29        
15 0 0 26        
16 17 6          
17 0 0          
18 15 1          
19 0 0          
20 21 0          
21 0 0          
22 19 1          
23 0 0          
24 31 9          
25 0 0          
26 26 1          
27 0 0          
28 33 1          
29 0 0          
30 34 1          
31 0 0          
32 45 14          
33 0 0          
34 37 1          
35 0 0          
36 52 3          
37 0 0          
38 49 3          
39 0 0          
40 63 12          
41 0 0          
42 64 4          
43 0 0          
44 72 1          
45 0 0          
46 70 1          
47 0 0          
48 91 23          

Table of A_4n