The Generating Functions for 2D Self-Avoiding Polygons of (Locally) Bounded Width The generating function (in the variable s) enumerating 2D (square-lattice) self-avoiding polygons of (local) width <= 1 equals 4 s - ------- 2 -1 + s The generating function (in the variable s) enumerating 2D (square-lattice) self-avoiding polygons of (local) width <= 2 equals 4 4 s (1 + s ) - --------------------- 4 2 6 -1 + s + 2 s + 2 s The generating function (in the variable s) enumerating 2D (square-lattice) self-avoiding polygons of (local) width <= 3 equals 4 10 8 6 14 2 4 12 s (4 s + 5 s + 2 s + s - s - s - 2 s + 1) - ------------------------------------------------------ 2 10 12 16 14 4 8 3 s + 6 s + 5 s + 2 s - 3 s + 2 s - 1 + 4 s The generating function (in the variable s) enumerating 2D (square-lattice) self-avoiding polygons of (local) width <= 4 equals 4 4 2 6 10 16 14 12 22 - s (1 - 7 s - 2 s + 8 s + 4 s - 187 s - 10 s + 16 s + 60 s 20 42 40 38 46 44 8 18 + 316 s - 34 s + 51 s - 76 s + 8 s - 6 s + 27 s - 92 s 36 34 28 26 32 30 24 / - 74 s + 172 s + 415 s + 260 s - 241 s - 350 s - 315 s ) / / 4 2 6 10 16 14 12 22 (-1 + 6 s + 4 s - 20 s + 70 s - 67 s - 28 s + 59 s + 408 s 20 42 40 38 46 44 8 18 - 133 s + 78 s - 123 s - 134 s - 8 s - 51 s - 18 s - 216 s 36 34 28 26 32 30 24 + 243 s - 358 s + 412 s - 414 s - 526 s + 634 s + 107 s 48 + 12 s ) The generating function (in the variable s) enumerating 2D (square-lattice) self-avoiding polygons of (local) width <= 5 equals 4 4 94 80 96 102 2 - s (1 - 15 s + 2740425 s - 2599783 s - 1589303 s - 1118441 s - 3 s 58 56 52 54 60 6 + 857061 s - 772539 s + 153578 s - 2021291 s + 1536221 s + 29 s 10 16 14 12 106 22 - 114 s + 2277 s + 553 s - 610 s - 401301 s + 8063 s 20 42 40 38 46 - 10181 s + 955636 s + 689206 s - 481760 s - 1355375 s 44 8 62 132 18 98 - 636018 s + 127 s + 880597 s + 132 s - 3066 s + 609687 s 84 82 128 100 124 + 1865281 s - 2414040 s - 111 s + 191774 s - 1783 s 126 122 92 90 120 130 + 1808 s - 373 s - 1576023 s - 1787340 s + 8817 s - 412 s 78 36 34 88 86 + 4772351 s - 458062 s + 172979 s + 2584196 s - 699594 s 114 76 116 64 66 + 86611 s + 478310 s + 24240 s - 3061059 s - 1336823 s 112 104 108 118 110 - 170947 s + 1081837 s - 8726 s - 35886 s + 138381 s 28 26 32 30 24 48 - 109805 s - 364 s + 232462 s - 48104 s + 40602 s + 316921 s 68 70 74 72 50 + 4059715 s + 1136940 s - 2743593 s - 1982540 s + 1814407 s ) / 4 94 80 96 102 / (-1 + 12 s - 2413950 s + 7087078 s + 4654730 s + 493635 s / 2 58 56 52 54 60 + 5 s - 964039 s - 3416579 s + 2744388 s + 104651 s + 1670896 s 6 10 16 14 12 106 - 60 s + 379 s - 2747 s - 1379 s + 754 s + 1715288 s 22 20 42 40 38 - 14140 s + 4203 s + 1040368 s - 777187 s - 663744 s 46 44 8 62 134 132 - 1005273 s + 1501235 s - 107 s + 1819936 s + 216 s - 648 s 18 98 84 82 128 + 3617 s - 2500732 s - 4055977 s - 4717325 s + 2804 s 100 124 126 122 92 + 514523 s - 798 s - 2478 s + 15062 s - 2785579 s 90 120 130 78 36 + 3408630 s - 57808 s - 250 s + 1140381 s + 284853 s 34 88 86 114 76 + 320955 s - 1034261 s + 3802981 s - 248550 s - 3484930 s 116 64 66 112 104 + 126060 s + 850061 s - 4563812 s + 265334 s - 1708055 s 108 118 110 28 26 - 647597 s + 33518 s - 80939 s + 29027 s + 59365 s 32 30 24 48 68 - 109896 s - 157086 s - 4346 s - 1977563 s - 1418630 s 70 74 72 50 + 6610043 s - 3508257 s + 949183 s + 596504 s )