The Generating Functions for 2D Self-Avoiding Walks of (Locally) Bounded Width The generating function (in the variable s) enumerating 2D (square-lattice) self-avoiding walks of (local) width <= 1 equals 4 3 2 (s - s - s + s + 2) s - --------------------------- 2 2 (2 s - 1) (s - 1) (s + 1) The generating function (in the variable s) enumerating 2D (square-lattice) self-avoiding walks of (local) width <= 2 equals 21 20 19 18 17 16 15 14 13 - (8 s - 2 s + 6 s - 10 s - 42 s - 9 s + 28 s + 84 s - 40 s 12 11 10 9 8 7 6 5 3 - 54 s - 18 s + 75 s + 30 s - 33 s - 32 s + 12 s + 16 s - 8 s 2 / 2 3 2 2 3 2 2 - s + 4 s + 2) s / ((s + s + 1) (2 s - s + 1) (2 s + s - 1) / 5 4 3 2 (2 s + s - 3 s - 2 s - s + 1) (s - 1)) The generating function (in the variable s) enumerating 2D (square-lattice) self-avoiding walks of (local) width <= 3 equals 59 58 57 56 55 54 53 52 (16 s - 24 s - 152 s + 180 s + 520 s - 470 s - 524 s + 508 s 51 50 49 48 47 46 45 - 1184 s - 162 s + 3183 s + 42 s - 223 s + 492 s - 9114 s 44 43 42 41 40 39 - 3420 s + 11141 s + 9276 s + 4485 s - 9760 s - 16707 s 38 37 36 35 34 33 - 4912 s + 10582 s + 18936 s + 1390 s - 12424 s - 8504 s 32 31 30 29 28 27 26 + 1504 s + 7126 s + 110 s - 790 s - 1004 s - 3241 s - 830 s 25 24 23 22 21 20 19 + 3023 s + 4954 s + 191 s - 3042 s - 2282 s - 572 s + 767 s 18 17 16 15 14 13 12 + 832 s + 859 s + 640 s - 223 s - 1200 s - 884 s + 318 s 11 10 9 8 7 6 5 4 + 858 s + 338 s - 264 s - 284 s - 46 s + 96 s + 66 s - 4 s 3 2 / 3 2 - 25 s - 8 s + 4 s + 2) s / ((2 s - s + s - 1) / 8 6 4 2 2 10 8 6 4 2 2 (s - s - 2 s + s - 1) (2 s - 7 s - s + 2 s + 3 s - 1) 4 3 (s - s + 1) 12 10 8 7 6 5 4 3 2 (2 s - 4 s + 9 s + 2 s - 5 s - 5 s + s + 4 s + 2 s + s - 1) (s + 1))