Illustration of initial terms:
. _ _ _ _
. _ _ _ _ _ _ _
. _ _ _ _ _ _  __
. _ _ _ _ _ _ _ _ _ 
. _ _ _ __ _     
. _ _         
. _ _ _ _ _ _
.
n: 1 2 3 4 5 6
a(n): 4 6 8 10 10 14
.
For n = 6 the diagram has 14 edges so a(6) = 14.
On the other hand the diagram has 13 vertices and two subparts or regions, so applying Euler's formula we have that a(6) = 13 + 2  1 = 14.
. _ _ _ _ _
. _ _ _ _ _ _ _ _ _ _
. _ _ _ _ _ _ _ _  _ _
. _ _ _ _  _ _ 
. _ _ _ _ __ _
. _ _ _ _   
.      
.      
.      
. _ _ _
.
n: 7 8 9
a(n): 12 14 16
.
For n = 9 the diagram has 16 edges so a(9) = 16.
On the other hand the diagram has 14 vertices and three subparts or regions, so applying Euler's formula we have that a(9) = 14 + 3  1 = 16.
Another way for the illustration of initial terms is as follows:

. n a(n) Diagram

_
1 4 _ _
_  _
2 6 _ _   _
_ __   _
3 8 _ _ _    _
_ _ _     _
4 10 _ _ _ __     _
_ _ _ _ _      _
5 10 _ _ _  _ _       _
_ _ _ _ __       _
6 14 _ _ _ _ _ _ _        _
_ _ _ _ _ _ _         _
7 12 _ _ _ _  _ _ __         _
_ _ _ _  _  _ _          _
8 14 _ _ _ _ _ _ _  _ _           _
_ _ _ _ _ _ __ _ __          
9 16 _ _ _ _ _  _ _ _ _ _         
_ _ _ _ _  _ _ _ _ _        
10 16 _ _ _ _ _ _  _ _  _ __      
_ _ _ _ _ _  _ _  _ _ _     
11 14 _ _ _ _ _ _  _ _ _  _ _ _    
_ _ _ _ _ _  _ _ __ _ _ __  
12 24 _ _ _ _ _ _ _  _ _ _ _  _ _ _ 
_ _ _ _ _ _ _  _  _ _  _ _ _
13 14 _ _ _ _ _ _ _   _ _ _ 
_ _ _ _ _ _ _  _ _ _ _
14 18 _ _ _ _ _ _ _ _  _ _ _
_ _ _ _ _ _ _ _  _ _
15 24 _ _ _ _ _ _ _ _  
_ _ _ _ _ _ _ _ 
16 22 _ _ _ _ _ _ _ _ _
...
