Each diagram gives what is counted by f(n,i,j) where N=6*n+j has the value stated. 
How to read the diagrams: the vertical edges have no weight. The diagonal edges have
weights indexed by two coordinates. The first coordinate is constant on vertical 
columns within a region of squares. The numbers to the right are the first 
coordinates of the columns suitably shifted: to get the actual weights one must 
alternately add either -2n+i or 2n+i starting from the left edge in the row. The 
second coordinate is constant on rows. The number to the left of a row is its second
coordinate, suitably shifted: to get the actual weight one must alternately add
either -2n+j or 2n+j starting from the top row.

N=16,17,18,19

     *
 1  / \   
   *   *   1,-1
-1  \ /
     *

N=20,21,22,23

       *
 1    / \        1,-1
     *   *
     |   |
     *   *
 5  / \ / \
   *   *   *  1,-5,5,-1  
-1  \ / \ /
     *   *

N=24,25,26,27

       *   *
 1    / \ / \ 
     *   *   *
-5    \ / \ /         1,-5,5,-1
       *   * 
 5    / \ / \
     *   *   *
     |   |   |
     *   *   *
 9  / \ / \ / \
   *   *   *   *    1,-9,5,-5,9,-1
-1  \ / \ / \ /
     *   *   *

N=28,29,30,31

        *   *
 1     / \ / \             1,-5,5,-1
      *   *   *
      |   |   |
      *   *   *
 5   / \ / \ / \
    *   *   *   *
-5   \ / \ / \ /         1,-9,5,-5,9,-1
      *   *   *
 9   / \ / \ / \
    *   *   *   *
    |   |   |   |
    *   *   *   *
13 / \ / \ / \ / \
  *   *   *   *   *  1,-13,5,-9,9,-5,13,1
-1 \ / \ / \ / \ /
    *   *   *   *

N=32,33,34,35

         *   *   *
  1     / \ / \ / \
       *   *   *   *
 -9     \ / \ / \ /              1,-9,5,-5,9,1
         *   *   *
  5     / \ / \ / \ 
       *   *   *   *
       |   |   |   |
       *   *   *   *
 9    / \ / \ / \ / \
     *   *   *   *   *
-5    \ / \ / \ / \ /       1,-13,5,-9,9,-5,13,-1
       *   *   *   *
13    / \ / \ / \ / \
     *   *   *   *   *
     |   |   |   |   |
     *   *   *   *   *
17  / \ / \ / \ / \ / \
   *   *   *   *   *   * 1,-17,5,-13,9,-9,13,-5,17,-1
-1  \ / \ / \ / \ / \ /
     *   *   *   *   *