The Generating Functions for 2D (2D site) Lattice-Animals of (Globally) Bounded Width The generating function (in the variable s) enumerating 2D (site) lattice animals of width <= 1 equals s - ------ -1 + s The generating function (in the variable s) enumerating 2D (site) lattice animals of width <= 2 equals 2 s (1 + s ) --------------------------- 3 2 (-1 + s + s + s ) (-1 + s) The generating function (in the variable s) enumerating 2D (site) lattice animals of width <= 3 equals 8 10 9 3 12 5 6 7 s (-1 + 2 s + 2 s - s - 2 s - 4 s + s + 4 s + 3 s - 2 s ) - ------------------------------------------------------------------ 10 9 8 6 4 3 3 2 (s - 3 s - s + 2 s + s - 4 s + 3 s - 1) (-1 + s + s + s ) The generating function (in the variable s) enumerating 2D (site) lattice animals of width <= 4 equals 8 34 33 10 9 2 3 17 s (1 - 4 s + 41 s + 2 s - 4 s - 35 s + 77 s + s + 11 s + 13 s 14 21 18 16 5 19 20 22 + 15 s + 6 s - 83 s + 75 s - 29 s - 113 s + 45 s + 99 s 23 4 6 7 11 12 24 28 + 4 s + 6 s - 25 s + 12 s - 48 s - 44 s - 108 s + 35 s 25 29 26 27 31 32 30 13 - 112 s - 40 s + 53 s + 24 s - 4 s + 17 s - 31 s - 63 s 15 / 8 10 9 2 3 17 + 94 s ) / ((-1 + 3 s + 10 s - 34 s - 29 s + 2 s - 4 s - 50 s / 14 21 18 16 5 19 20 22 23 + 41 s - 4 s - s - 30 s + 5 s + 7 s + 18 s - 15 s - 14 s 4 6 7 11 12 24 26 13 15 - 13 s + 10 s + 17 s - 35 s + 23 s + 5 s + s + 47 s - 7 s ) 10 9 8 6 4 3 (s - 3 s - s + 2 s + s - 4 s + 3 s - 1)) The generating function (in the variable s) enumerating 2D (site) lattice animals of width <= 5 equals 8 90 95 87 86 - s (1 - 5 s + 1692 s - 197863 s + 47455 s + 598283 s - 439245 s 101 10 9 104 107 2 103 97 + 625 s - 3388 s + 1278 s + s - s - 2 s + 53 s - 11485 s 100 99 98 96 93 94 + 799 s - 2536 s - 9079 s + 8342 s + 20860 s + 68356 s 3 108 27 17 14 78 + 27 s + s - 112636 s - 15774 s + 11681 s - 10914107 s 80 79 75 76 105 + 1067199 s - 7073922 s + 15711884 s + 9849651 s - 21 s 36 21 102 106 74 32 - 4471707 s - 31280 s + 195 s - 3 s + 6483175 s + 1518289 s 18 33 16 30 66 + 4375 s + 2883846 s - 19476 s - 783978 s + 1741480 s 70 34 84 82 47 + 9090323 s + 1729412 s - 1831506 s + 3685599 s + 8314302 s 83 67 5 19 20 55 + 202594 s - 12627804 s - 139 s + 23588 s + 11472 s - 4651193 s 63 22 23 4 6 7 - 493779 s - 44561 s + 44607 s + 58 s - 391 s + 134 s 35 31 29 65 11 - 1714804 s - 303357 s - 326327 s + 19488386 s - 6589 s 12 49 48 59 24 + 417 s + 12030822 s + 16381163 s + 17190676 s + 147141 s 25 26 13 15 38 + 75717 s - 105647 s + 12339 s - 4485 s - 1492703 s 72 71 73 51 57 - 11255183 s - 395711 s - 8018941 s - 10938695 s + 2771510 s 60 69 61 39 54 + 425821 s + 3886477 s - 19948764 s + 1951561 s + 686972 s 50 68 53 28 52 - 1579627 s - 9541669 s - 418239 s - 31357 s - 8031479 s 41 77 46 45 42 + 7108644 s - 3619826 s - 4213322 s - 11530662 s + 5036989 s 43 44 81 56 64 - 1986315 s - 9781162 s + 5226562 s - 6115256 s + 19890478 s 37 40 62 58 89 - 4183786 s + 5156296 s - 21070312 s + 15751111 s + 223569 s 88 92 91 85 / + 717413 s - 109551 s - 255417 s - 1691178 s ) / ((1 - 4 s / 8 10 9 2 3 27 17 + 552 s - 207 s + 1005 s - 4 s + 14 s - 24906 s + 4340 s 14 78 80 79 75 76 36 + 2246 s + 149 s - 13 s + 14 s - 1172 s + 305 s + 269472 s 21 74 32 18 33 16 + 2558 s - 3722 s - 50294 s - 1820 s - 140491 s + 2303 s 30 66 70 34 82 47 83 + 50454 s - 42300 s + 18467 s - 70612 s - 4 s + 361257 s + s 67 5 19 20 55 63 - 66723 s - 27 s - 16255 s - 19286 s - 259843 s + 153686 s 22 23 4 6 7 35 31 + 29293 s + 31919 s + 53 s - 236 s - 174 s + 135892 s + 49705 s 29 65 11 12 49 48 - 6770 s + 34159 s - 2056 s - 1672 s - 132371 s + 213578 s 59 24 25 26 13 15 - 170827 s + 5856 s - 17489 s - 21168 s + 938 s + 1597 s 38 72 71 73 51 57 - 40985 s + 3738 s + 14389 s - 3175 s - 34221 s + 32615 s 60 69 61 39 54 50 - 182273 s + 1407 s - 42468 s - 207702 s - 144942 s - 275944 s 68 53 28 52 41 77 - 37794 s + 151532 s - 28836 s + 236030 s - 155376 s + 444 s 46 45 42 43 44 81 + 213996 s - 41147 s - 180217 s - 249875 s - 223667 s - 11 s 56 64 37 40 62 - 109936 s + 116789 s + 189139 s - 211492 s + 104062 s 58 8 10 9 2 3 17 - 25672 s ) (-1 + 3 s + 10 s - 34 s - 29 s + 2 s - 4 s - 50 s 14 21 18 16 5 19 20 22 23 + 41 s - 4 s - s - 30 s + 5 s + 7 s + 18 s - 15 s - 14 s 4 6 7 11 12 24 26 13 15 - 13 s + 10 s + 17 s - 35 s + 23 s + 5 s + s + 47 s - 7 s ))