There is initially one region and the 1-gon and 2-gon are ignored, so a(1) and a(2) result in one region. Each line of the n-gon should cross as many lines as possible and avoid intersecting previous intersections.

For n = 3 the plane is empty, so the trigon can only create 1 extra region. Thus a(3) = 2.

For n = 4 each tetragon edge intersects a maximum of 2 trigon edges, creating a total of 4 new regions. Two trigon edges intersect 2 tetragon edges, adding 4 regions, and the last trigon edge intersects all 4 tetragon edges, adding another 4 regions. Thus a(4) = 2 + 4 + 4 = 10.