For words avoiding the pattern, [0, 0, 0], with spacings <=, 0 2 t + t + 1 the generating function is, - --------------- 2 3 t + t - 1 + t the first , 30, terms are , [2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136, 5768, 10609, 19513, 35890, 66012, 121415, 223317, 410744, 755476, 1389537, 2555757, 4700770, 8646064, 15902591, 29249425, 53798080, 98950096] the asympt. constant is, 1.839286755 For words avoiding the pattern, [1, 0, 0], with spacings <=, 0 1 the generating function is, ------------ 3 1 - 2 t + t the first , 30, terms are , [2, 4, 7, 12, 20, 33, 54, 88, 143, 232, 376, 609, 986, 1596, 2583, 4180, 6764, 10945, 17710, 28656, 46367, 75024, 121392, 196417, 317810, 514228, 832039, 1346268, 2178308, 3524577] the asympt. constant is, 1.618033989 For words avoiding the pattern, [0, 1, 0], with spacings <=, 0 2 t + 1 the generating function is, - ----------------- 3 2 t - 1 + 2 t - t the first , 30, terms are , [2, 4, 7, 12, 21, 37, 65, 114, 200, 351, 616, 1081, 1897, 3329, 5842, 10252, 17991, 31572, 55405, 97229, 170625, 299426, 525456, 922111, 1618192, 2839729, 4983377, 8745217, 15346786, 26931732] the asympt. constant is, 1.754877666 The low champion is, [1, 0, 0], with asympt. constant, 1.618033989 The high champion is, [0, 0, 0], with asympt. constant, 1.839286755 For words avoiding the pattern, [0, 0, 0], with spacings <=, 1 2 4 3 5 t + 2 t + 2 t + 1 + t + t the generating function is, - ----------------------------- 3 6 5 2 t - 1 + t + t + t the first , 30, terms are , [2, 4, 7, 13, 23, 40, 72, 129, 229, 409, 730, 1300, 2319, 4137, 7375, 13152, 23456, 41825, 74585, 133009, 237186, 422964, 754263, 1345045, 2398567, 4277288, 7627528, 13601889, 24255773, 43254441] the asympt. constant is, 1.783263984 For words avoiding the pattern, [1, 0, 0], with spacings <=, 1 1 the generating function is, - ---------------------- 3 5 6 2 t - t - 1 - t + t the first , 30, terms are , [2, 4, 7, 12, 19, 30, 46, 70, 105, 157, 233, 345, 509, 750, 1103, 1621, 2380, 3493, 5124, 7515, 11019, 16155, 23682, 34714, 50882, 74578, 109306, 160203, 234796, 344118] the asympt. constant is, 1.465571232 For words avoiding the pattern, [0, 1, 0], with spacings <=, 1 7 2 4 3 5 6 t + 1 + t + 2 t + 4 t + 2 t + 3 t + 2 t the generating function is, - --------------------------------------------- 6 5 3 8 7 4 t + t - 1 + t + t + t + t - t the first , 30, terms are , [2, 4, 7, 12, 18, 26, 39, 59, 92, 146, 229, 357, 553, 852, 1316, 2038, 3161, 4910, 7623, 11824, 18332, 28412, 44038, 68278, 105876, 164191, 254622, 394826, 612201, 949236] the asympt. constant is, 1.550586419 The low champion is, [1, 0, 0], with asympt. constant, 1.465571232 The high champion is, [0, 0, 0], with asympt. constant, 1.783263984 For words avoiding the pattern, [0, 0, 0], with spacings <=, 2 6 5 10 7 9 the generating function is, - (-1 - t - 7 t - 5 t + 7 t - 4 t + 5 t 11 2 12 13 3 4 / + 6 t - 2 t + 3 t + t - 3 t - 4 t ) / ( / 14 13 12 4 11 9 8 7 5 t + 2 t + 3 t - 2 t + 2 t - 2 t - 2 t - 3 t - t - t + 1) the first , 30, terms are , [2, 4, 7, 13, 23, 40, 65, 106, 178, 307, 530, 906, 1543, 2625, 4463, 7578, 12877, 21921, 37354, 63630, 108310, 184311, 313656, 533812, 908514, 1546311, 2632036, 4480175, 7625736, 12979279] the asympt. constant is, 1.702057535 For words avoiding the pattern, [1, 0, 0], with spacings <=, 2 2 3 5 6 4 1 + t + t + t + t + t + t the generating function is, - ------------------------------ 2 5 7 9 t - t - t + t - 1 + t the first , 30, terms are , [2, 4, 7, 12, 19, 30, 44, 65, 94, 135, 191, 270, 378, 529, 737, 1025, 1422, 1972, 2730, 3778, 5224, 7221, 9977, 13783, 19035, 26286, 36294, 50109, 69177, 95498] the asympt. constant is, 1.380277569 For words avoiding the pattern, [0, 1, 0], with spacings <=, 2 the generating function is, 11 12 5 7 6 8 9 2 10 4 t + t + t + 2 t - t + 2 t + 2 t - 1 - t + 2 t - 2 t - ---------------------------------------------------------------- 5 7 11 13 6 3 2 4 -2 t - 2 t + t + t + 2 t + 1 - 2 t - t + t + 2 t the first , 30, terms are , [2, 4, 7, 12, 18, 26, 36, 48, 66, 92, 132, 195, 289, 430, 634, 924, 1338, 1924, 2764, 3978, 5740, 8316, 12075, 17553, 25518, 37055, 53749, 77881, 112772, 163280] the asympt. constant is, 1.449389380 The low champion is, [1, 0, 0], with asympt. constant, 1.380277569 The high champion is, [0, 0, 0], with asympt. constant, 1.702057535 For words avoiding the pattern, [0, 0, 0], with spacings <=, 3 14 24 26 25 27 the generating function is, - (1 - 29 t + t + 9 t + 2 t + 6 t + t 15 16 19 9 11 10 12 13 2 - 33 t - 20 t + 5 t + 4 t - 4 t + 4 t - 16 t - 24 t + 2 t 3 4 6 5 7 8 17 18 20 + 3 t + 6 t + 11 t + 7 t + 10 t + 14 t - 13 t - 4 t + 8 t 22 23 21 / 28 27 26 25 24 23 + 13 t + 11 t + 9 t ) / (t + t + 4 t + 3 t + 3 t + 3 t / 21 19 18 17 16 15 14 12 11 + 2 t - 6 t - 9 t - 5 t - 13 t - 3 t - 3 t + 2 t + 3 t 10 9 7 5 - t + 8 t + 3 t + 3 t + t - 1) the first , 30, terms are , [2, 4, 7, 13, 23, 40, 65, 106, 169, 278, 466, 774, 1301, 2171, 3631, 6060, 10088, 16775, 27875, 46407, 77243, 128698, 214391, 357098, 594877, 990678, 1649879, 2747344, 4575037, 7618775] the asympt. constant is, 1.665364762 For words avoiding the pattern, [1, 0, 0], with spacings <=, 3 11 10 9 6 5 4 3 2 t + 2 t + t - t - t - t - t - t - t - 1 the generating function is, - ------------------------------------------------- 2 3 4 6 11 9 3 (t - 1 - t + t + t - t + t - t ) (-1 + t ) the first , 30, terms are , [2, 4, 7, 12, 19, 30, 44, 65, 92, 129, 179, 248, 338, 460, 621, 837, 1123, 1505, 2010, 2683, 3573, 4755, 6321, 8400, 11152, 14802, 19636, 26044, 34532, 45781] the asympt. constant is, 1.324717957 For words avoiding the pattern, [0, 1, 0], with spacings <=, 3 5 7 6 8 15 2 the generating function is, - (-3 t - 2 t - 4 t - 3 t - 1 + 9 t - 2 t 21 3 4 11 13 18 17 16 22 - t + 3 t - 2 t - 4 t + 6 t + 9 t + 6 t + 7 t + 10 t + 2 t 14 20 19 12 9 23 10 / 4 8 + 10 t + 4 t + 5 t + 8 t + 2 t + t + 4 t ) / (t + 1 + 2 t / 9 12 11 20 19 18 24 17 15 23 22 - 2 t - t - t + t + t + t + t + 3 t + 2 t + t + t 21 5 3 10 6 7 + t - t - t - t - t + t - t ) the first , 30, terms are , [2, 4, 7, 12, 18, 26, 36, 48, 61, 77, 100, 130, 176, 245, 345, 493, 705, 1002, 1404, 1950, 2681, 3655, 4978, 6771, 9232, 12645, 17402, 24045, 33309, 46217] the asympt. constant is, 1.379128044 The low champion is, [1, 0, 0], with asympt. constant, 1.324717957 The high champion is, [0, 0, 0], with asympt. constant, 1.665364762 For words avoiding the pattern, [0, 0, 0], with spacings <=, 4 21 22 23 27 the generating function is, - (1 - 85 t + 27 t + 120 t + 213 t 25 26 24 55 57 58 15 54 + 231 t + 272 t + 183 t + t + 2 t + t + t - 91 t + 2 t 52 53 51 56 9 11 10 12 2 - t - t - 3 t + 2 t + 22 t + 22 t + 37 t + 4 t + 2 t 3 48 4 6 5 7 42 16 19 + 3 t - 16 t + 6 t + 14 t + 10 t + 16 t + 20 t - 91 t - 165 t 8 18 17 32 33 34 20 + 23 t - 177 t - 145 t - 83 t - 119 t - 109 t - 127 t 30 29 28 31 14 38 39 40 + 27 t + 117 t + 183 t - 60 t - 43 t + 7 t + 13 t + 17 t 44 43 41 36 35 37 47 46 - 10 t - 4 t + 16 t - 29 t - 79 t - 26 t - 31 t - 30 t 45 13 49 50 / 21 22 23 - 23 t - 4 t - 15 t - 8 t ) / (-1 - 20 t - 47 t - 39 t / 27 25 26 24 59 57 15 54 52 + 71 t + 7 t + 7 t - 8 t + t + t + t + 7 t - 2 t - t 51 9 11 10 12 48 6 7 42 - 5 t + 8 t + 11 t - 4 t + 7 t - 15 t + 3 t + 3 t - 5 t 16 19 18 17 32 33 34 20 - 24 t - 27 t - 8 t + t + 7 t + 21 t - 16 t - 45 t 30 29 28 31 14 38 39 40 + 59 t + 58 t + 47 t + 41 t + 10 t - 25 t - t + 2 t 44 43 41 36 35 37 46 45 13 + 2 t + 10 t + 4 t - 42 t - 29 t - 9 t + 2 t + 6 t - 2 t 49 50 - 3 t - 7 t ) the first , 30, terms are , [2, 4, 7, 13, 23, 40, 65, 106, 169, 278, 443, 705, 1148, 1861, 3034, 5004, 8236, 13457, 22054, 35982, 58596, 95600, 155885, 254196, 414990, 677237, 1105047, 1804675, 2946080, 4809057] the asympt. constant is, 1.632273568 For words avoiding the pattern, [1, 0, 0], with spacings <=, 4 5 3 14 17 4 16 the generating function is, - (-6 t - 1 - 4 t + 11 t + t - 5 t + 4 t 6 15 13 9 12 10 8 2 7 - 7 t + 8 t + 9 t - 6 t + 4 t - 4 t - 7 t - 3 t - 2 t - 7 t ) / 13 10 9 8 2 3 16 15 18 11 19 / (1 - t + t + t + t - t - t - t - 2 t + t + t + t / 14 4 7 - t - t + t ) the first , 30, terms are , [2, 4, 7, 12, 19, 30, 44, 65, 92, 129, 175, 238, 316, 422, 559, 738, 965, 1262, 1640, 2133, 2764, 3581, 4628, 5982, 7716, 9951, 12820, 16515, 21260, 27370] the asympt. constant is, 1.285199033 For words avoiding the pattern, [0, 1, 0], with spacings <=, 4 15 30 25 9 11 10 the generating function is, - (1 - t + t + 2 t - 5 t - 5 t + t 12 2 4 6 5 7 16 29 19 8 18 17 - 2 t + t + 2 t + t - t - 2 t - t + t + 4 t + t - t + t 22 23 24 14 27 28 26 13 21 + 2 t + 2 t + 2 t - 2 t + 2 t + 2 t + 2 t - 2 t + 4 t ) / 12 31 29 18 2 15 11 21 10 / (-t + 2 t - 1 + t + t + 2 t - t - t + 4 t - 3 t - 4 t / 19 8 7 4 22 20 9 17 6 5 - 4 t - 3 t + 2 t - 2 t + 2 t + 4 t + 4 t - 2 t - 2 t + 2 t 3 + t ) the first , 30, terms are , [2, 4, 7, 12, 18, 26, 36, 48, 61, 77, 95, 115, 144, 180, 232, 309, 418, 576, 800, 1111, 1528, 2092, 2824, 3772, 5007, 6583, 8634, 11308, 14823, 19494] the asympt. constant is, 1.332023989 The low champion is, [1, 0, 0], with asympt. constant, 1.285199033 The high champion is, [0, 0, 0], with asympt. constant, 1.632273568 For words avoiding the pattern, [0, 0, 0], with spacings <=, 5 94 88 51 53 the generating function is, - (-1 + 6 t - t + 136 t + 6118 t + 1577 t 87 89 85 84 83 82 81 + 177 t + 152 t + 330 t + 268 t + 459 t + 313 t + 216 t 98 32 79 78 80 49 56 - 4 t - 2462 t - 395 t - 390 t + 19 t + 6912 t - 2789 t 66 48 47 70 73 72 77 - 3316 t + 5636 t + 3512 t - 1894 t - 127 t - 299 t - 650 t 76 46 9 10 107 2 3 63 - 429 t + 1156 t - 31 t - 55 t + t - 2 t - 3 t + 593 t 101 60 62 64 59 58 61 + 11 t + 1033 t + 1477 t - 1022 t + 292 t - 1056 t + 1707 t 50 5 18 7 42 39 52 + 7238 t - 10 t + 403 t - 19 t - 1805 t + 829 t + 3910 t 19 65 8 106 17 11 41 + 450 t - 2543 t - 29 t + 3 t + 201 t - 52 t - 806 t 36 38 111 12 13 45 110 105 + 386 t + 1456 t + t - 61 t - 36 t - 805 t - t + 3 t 102 103 69 90 75 71 55 - 5 t + 3 t - 2789 t + 137 t - 186 t - 878 t - 2118 t 54 57 97 99 74 15 14 40 - 437 t - 2566 t + 2 t + 26 t + 24 t + 35 t - 7 t + 16 t 91 93 35 34 37 95 100 + 64 t - 56 t - 351 t - 1002 t + 1136 t - 38 t - 15 t 67 68 4 6 30 96 86 16 - 3667 t - 3785 t - 6 t - 17 t - 2377 t - 4 t + 207 t + 74 t 20 21 22 23 25 26 24 + 579 t + 786 t + 813 t + 772 t + 333 t - 139 t + 613 t 27 28 29 33 31 44 - 779 t - 1325 t - 1981 t - 1809 t - 2491 t - 2134 t 43 92 / 94 88 51 53 - 2374 t + 42 t ) / (1 - 21 t - t + 20 t + 1119 t + 1575 t / 112 87 89 85 84 83 82 81 + t + 96 t + 5 t - 2 t + 183 t - 125 t - 100 t - 89 t 98 32 79 78 80 49 56 + 10 t - 291 t - 3 t - 92 t - 263 t - 301 t + 664 t 66 48 47 70 73 72 77 + 114 t - 543 t - 694 t - 848 t - 190 t - 678 t + 96 t 76 46 9 10 107 104 63 101 + 166 t - 674 t - 8 t + 4 t + 2 t + 4 t + 171 t - 24 t 60 62 64 59 58 61 50 - 393 t - 13 t + 792 t - 825 t - 843 t - 662 t + 400 t 18 7 42 39 52 19 65 106 - 29 t - 6 t + 393 t + 237 t + 1568 t + 60 t + 682 t + t 17 11 41 36 38 111 12 13 - 13 t - 11 t + 239 t - 511 t - 311 t - 2 t - t - 14 t 45 110 105 102 103 69 108 90 - 465 t + t - 5 t + 17 t - 8 t - 902 t + t - t 75 71 55 54 57 97 99 + 74 t - 854 t + 1142 t + 1548 t - 66 t - 14 t - 35 t 74 15 14 40 91 93 35 34 - 380 t - 2 t - 5 t + 327 t + 58 t + 64 t - 446 t - 551 t 37 95 100 67 68 30 96 - 455 t + 20 t + 42 t + 110 t - 238 t - 97 t - 17 t 86 16 20 21 22 23 25 26 + 152 t + 10 t + 47 t - 9 t + 62 t + 106 t + 120 t + 145 t 24 27 28 29 33 31 44 + 132 t + 203 t + 85 t + 126 t - 539 t - 121 t + 126 t 43 92 + 329 t + 32 t ) the first , 30, terms are , [2, 4, 7, 13, 23, 40, 65, 106, 169, 278, 443, 705, 1117, 1760, 2807, 4485, 7221, 11575, 18815, 30401, 48976, 78922, 127053, 204529, 328383, 528383, 849390, 1366728, 2197264, 3533774] the asympt. constant is, 1.608622549 For words avoiding the pattern, [1, 0, 0], with spacings <=, 5 15 9 10 2 3 the generating function is, - (-1 - 2 t + 4 t - 25 t - 27 t - 4 t - 6 t 11 16 5 18 7 14 12 13 - 25 t + 11 t - 12 t + 16 t - 19 t - 6 t - 23 t - 14 t 17 4 19 8 21 6 20 / 3 4 + 16 t - 9 t + 12 t - 23 t + 2 t - 16 t + 6 t ) / (-t - t / 10 6 14 11 16 9 13 23 22 17 + t - t + t + 2 t - t + t + 2 t + t + t + 1 - 2 t 19 18 20 12 5 24 - 2 t - 2 t - t + t - t + t ) the first , 30, terms are , [2, 4, 7, 12, 19, 30, 44, 65, 92, 129, 175, 238, 313, 413, 540, 703, 908, 1173, 1504, 1926, 2452, 3118, 3954, 5015, 6343, 8020, 10126, 12775, 16098, 20281] the asympt. constant is, 1.255422871 For words avoiding the pattern, [0, 1, 0], with spacings <=, 5 39 43 40 41 9 the generating function is, - (1 + t + 9 t + 5 t + 8 t + 7 t - 2 t 10 44 45 46 2 3 5 18 7 - t + 4 t + 3 t + 2 t + 2 t + 2 t + 3 t - 14 t + 2 t 29 28 27 26 25 19 8 17 + 14 t + 12 t + 13 t + 8 t + 7 t - 13 t + 3 t - 13 t 11 12 13 47 15 14 4 6 42 - 6 t - 4 t - 11 t + t - 14 t - 12 t + 4 t + 4 t + 6 t 31 32 33 35 34 36 37 38 + 18 t + 20 t + 21 t + 20 t + 22 t + 20 t + 15 t + 12 t 30 16 20 21 22 23 24 / + 16 t - 12 t - 14 t - 9 t - 10 t - 3 t - 3 t ) / (-1 + t / 39 43 40 41 9 10 44 45 46 3 5 7 + 3 t + t + t + t + 2 t - 2 t + t + t + t + t + t + t 29 28 27 26 25 19 8 11 12 - t + 2 t - 2 t + 2 t - 5 t - 2 t - 2 t + 2 t - 3 t 13 47 48 15 14 4 6 42 31 32 35 + 3 t + t + t + 2 t + t - t - t + t - t - t + 3 t 36 37 38 30 21 22 23 24 + t + 5 t + 3 t - t - 5 t + t - 5 t + 2 t ) the first , 30, terms are , [2, 4, 7, 12, 18, 26, 36, 48, 61, 77, 95, 115, 137, 161, 194, 236, 295, 377, 493, 658, 890, 1210, 1636, 2208, 2950, 3909, 5118, 6632, 8527, 10888] the asympt. constant is, 1.295437148 The low champion is, [1, 0, 0], with asympt. constant, 1.255422871 The high champion is, [0, 0, 0], with asympt. constant, 1.608622549 This took, 154.213, seconds