There are 6 ways to form a rectangle from 3 rectangles with same area:
++ ++++ ++ +++ +++ +++
                
++     +++         
             ++ ++ 
++        +++      
                
++ ++++ +++ ++ +++ +++
So a(3)=6.
b(n) in the given formula is the sum of the appropriate tilings from certain 'frames'. A number that appears in a subrectangle in a frame is the number of rectangles into which the subrectangle is to be divided. Tilings are also counted that are from a reflection and/or halfturn of the frame.
For n = 6 there are 3(X2) frames:
++++ +++ +++
         
     +++   2 
+++         
      ++   +++
  +++        
    +++  +++ 
         
++++ +++ +++
2 ways 2 ways 8 ways
The only other frames which yield desired tilings are obtained by rotating each frame above by 90 degrees and scaling it to fit a rectangle with the inverse aspect ratio.
So b(6) = 2(2+2+8) = 24, and a(6) = b(6)+4*a(5)+2*a(4)4*a(3)2*a(2) = 24+4*88+2*214*62*2 = 390.
For n = 7 we can use 7(X2) frames:
+++
  
  
 4 3 
  
  
  
+++
63 ways [of creating tilings counted by b(7)]
+++ +++ +++ +++ +++ +++
         +++    ++++ 
 +++  +++ 2  2     +++    
 3   2   +++   2       
     2      3       +++
      +++   +++2   
+++ +++   +++    +++
+++ +++ +++ +++ +++ +++
24 ways 16 ways 12 ways 10 ways 8 ways 4 ways
As for n = 6, these are only half the frames and tilings.
So b(7) = 2(63+24+16+12+10+8+4) = 274, and a(7) = b(7)+4*a(6)+2*a(5)4*a(4)2*a(3) = 274+4*390+2*884*212*6 = 1914.
(End)
