The Generating Functions for

             2D Self-Avoiding Polygons of (Globally) Bounded Width

The generating function (in the variable s) 
 enumerating  2D self-avoiding polygons 
of width <=   1   equals 
                                        4
                                       s
                               - ---------------
                                 (s - 1) (s + 1)



The generating function (in the variable s) 
 enumerating  2D self-avoiding polygons 
of width <=   2   equals 
                               6      4    2       4
                             (s  - 3 s  + s  - 1) s
                     - -----------------------------------
                           2        4
                       (2 s  - 1) (s  + 1) (s - 1) (s + 1)



The generating function (in the variable s) 
 enumerating  2D self-avoiding polygons 
of width <=   3   equals 
      24      22      20      18       16       14       12       10      8
- (4 s   - 6 s   + 5 s   + 9 s   - 13 s   + 28 s   - 17 s   - 12 s   + 3 s

        6      4      2       4   /
     - s  + 7 s  - 4 s  + 1) s   /  ((
                                /

       20      18      16      14    12      10      8      6      4      2
    4 s   - 2 s   - 5 s   + 8 s   - s   + 2 s   - 4 s  + 2 s  + 3 s  - 4 s  + 1

          2        4
    ) (2 s  - 1) (s  + 1))



The generating function (in the variable s) 
 enumerating  2D self-avoiding polygons 
of width <=   4   equals 
     68      66         64         62         60         58         56
(96 s   + 4 s   - 1166 s   + 1535 s   + 1917 s   - 6156 s   + 2657 s

             54         52         50         48         46          44
     + 6394 s   - 6823 s   - 3717 s   + 7064 s   + 5704 s   - 13808 s

             42          40         38          36          34         32
     - 1454 s   + 18488 s   - 9170 s   - 13073 s   + 15829 s   + 1985 s

              30         28         26         24        22         20
     - 13376 s   + 5609 s   + 5313 s   - 4849 s   - 756 s   + 2466 s

            18        16        14       12        10       8      6       4
     - 384 s   - 978 s   + 459 s   + 78 s   - 137 s   + 64 s  + 8 s  - 22 s

          2       4   /      54       50       48        46        44        42
     + 8 s  - 1) s   /  ((4 s   - 39 s   - 33 s   + 324 s   - 351 s   - 264 s
                    /

            40        38        36        34       32        30        28
     + 871 s   - 402 s   - 535 s   + 714 s   + 51 s   - 672 s   + 379 s

            26        24       22        20        18       16        14
     + 270 s   - 390 s   + 43 s   + 182 s   - 132 s   - 76 s   + 133 s

           12       10       8       6      4      2
     - 71 s   - 15 s   + 40 s  - 12 s  - 8 s  + 6 s  - 1) (

       20      18      16      14    12      10      8      6      4      2
    4 s   - 2 s   - 5 s   + 8 s   - s   + 2 s   - 4 s  + 2 s  + 3 s  - 4 s  + 1

    ))