For words avoiding the pattern, [0, 1, 1, 0], with spacings <=, 0 3 t + 1 the generating function is, - ----------------- 4 3 t - t - 1 + 2 t the first , 30, terms are , [2, 4, 8, 15, 28, 52, 97, 181, 338, 631, 1178, 2199, 4105, 7663, 14305, 26704, 49850, 93058, 173717, 324288, 605368, 1130077, 2109583, 3938086, 7351463, 13723420, 25618337, 47823297, 89274637, 166654357] the asympt. constant is, 1.866760399 For words avoiding the pattern, [0, 0, 0, 0], with spacings <=, 0 3 2 t + t + t + 1 the generating function is, - -------------------- 3 4 2 t + t + t - 1 + t the first , 30, terms are , [2, 4, 8, 15, 29, 56, 108, 208, 401, 773, 1490, 2872, 5536, 10671, 20569, 39648, 76424, 147312, 283953, 547337, 1055026, 2033628, 3919944, 7555935, 14564533, 28074040, 54114452, 104308960, 201061985, 387559437] the asympt. constant is, 1.927561975 For words avoiding the pattern, [1, 0, 0, 0], with spacings <=, 0 1 the generating function is, ------------ 4 t + 1 - 2 t the first , 30, terms are , [2, 4, 8, 15, 28, 52, 96, 177, 326, 600, 1104, 2031, 3736, 6872, 12640, 23249, 42762, 78652, 144664, 266079, 489396, 900140, 1655616, 3045153, 5600910, 10301680, 18947744, 34850335, 64099760, 117897840] the asympt. constant is, 1.839286755 For words avoiding the pattern, [0, 1, 0, 0], with spacings <=, 0 3 t + 1 the generating function is, - ----------------- 4 3 t - t - 1 + 2 t the first , 30, terms are , [2, 4, 8, 15, 28, 52, 97, 181, 338, 631, 1178, 2199, 4105, 7663, 14305, 26704, 49850, 93058, 173717, 324288, 605368, 1130077, 2109583, 3938086, 7351463, 13723420, 25618337, 47823297, 89274637, 166654357] the asympt. constant is, 1.866760399 For words avoiding the pattern, [1, 1, 0, 0], with spacings <=, 0 1 the generating function is, ------------ 4 t + 1 - 2 t the first , 30, terms are , [2, 4, 8, 15, 28, 52, 96, 177, 326, 600, 1104, 2031, 3736, 6872, 12640, 23249, 42762, 78652, 144664, 266079, 489396, 900140, 1655616, 3045153, 5600910, 10301680, 18947744, 34850335, 64099760, 117897840] the asympt. constant is, 1.839286755 For words avoiding the pattern, [1, 0, 1, 0], with spacings <=, 0 2 t + 1 the generating function is, ------------------------- 3 4 2 -2 t + t + 1 - 2 t + t the first , 30, terms are , [2, 4, 8, 15, 28, 53, 100, 188, 354, 667, 1256, 2365, 4454, 8388, 15796, 29747, 56020, 105497, 198672, 374140, 704582, 1326871, 2498768, 4705689, 8861770, 16688516, 31427872, 59185079, 111457548, 209897245] the asympt. constant is, 1.883203506 The low champion is, [1, 0, 0, 0], with asympt. constant, 1.839286755 The high champion is, [0, 0, 0, 0], with asympt. constant, 1.927561975 For words avoiding the pattern, [0, 1, 1, 0], with spacings <=, 1 the generating function is, 3 8 9 7 5 11 12 6 -1 - t - 2 t + t + t - 2 t - t + t - 4 t - ------------------------------------------------------------------------ 3 12 9 13 5 8 7 10 4 11 -2 t + t - 2 t - 3 t + t + 2 t + 3 t - 4 t + t + 1 - t + t the first , 30, terms are , [2, 4, 8, 15, 28, 52, 92, 162, 282, 493, 858, 1498, 2621, 4592, 8055, 14131, 24804, 43524, 76373, 133984, 235036, 412270, 723120, 1268355, 2224682, 3902151, 6844499, 12005621, 21058567, 36938087] the asympt. constant is, 1.754060167 For words avoiding the pattern, [0, 0, 0, 0], with spacings <=, 1 the generating function is, 5 6 7 3 2 4 8 9 11 10 1 + t + 3 t + 5 t + 2 t + 2 t + 2 t + 3 t - t - t + t + t - --------------------------------------------------------------------- 10 12 8 7 3 5 -t + t + 2 t + 3 t + 2 t + 3 t - 1 + t the first , 30, terms are , [2, 4, 8, 15, 29, 56, 103, 192, 364, 689, 1301, 2452, 4626, 8730, 16462, 31052, 58589, 110536, 208527, 393376, 742109, 1400013, 2641128, 4982520, 9399624, 17732538, 33452672, 63108899, 119055854, 224600640] the asympt. constant is, 1.886514351 For words avoiding the pattern, [1, 0, 0, 0], with spacings <=, 1 the generating function is, 4 3 2 t - t + t - t + 1 - -------------------------------------------------- 5 7 3 8 4 6 2 3 t - t + 3 t + t - 4 t - t + 3 t - 3 t - 1 the first , 30, terms are , [2, 4, 8, 15, 28, 52, 94, 169, 304, 544, 972, 1737, 3100, 5530, 9866, 17597, 31382, 55968, 99810, 177989, 317408, 566028, 1009378, 1799995, 3209874, 5724052, 10207504, 18202685, 32460190, 57885096] the asympt. constant is, 1.783263984 For words avoiding the pattern, [0, 1, 0, 0], with spacings <=, 1 the generating function is, - ( 7 3 2 4 8 9 11 5 6 10 1 + t + 3 t + 2 t + t + 2 t + 2 t + 2 t + 2 t + 3 t + 7 t + 2 t / 8 7 4 2 6 12 ) / (t + t + 4 t - 1 + t + t - 2 t + 2 t ) / the first , 30, terms are , [2, 4, 8, 15, 28, 52, 91, 161, 284, 505, 894, 1585, 2821, 5013, 8911, 15826, 28130, 49984, 88799, 157774, 280322, 498095, 884959, 1572376, 2793767, 4963896, 8819695, 15670491, 27842997, 49470531] the asympt. constant is, 1.776769863 For words avoiding the pattern, [1, 1, 0, 0], with spacings <=, 1 the generating function is, 8 5 t - 2 t - 1 ------------------------------------------------------------- 12 10 9 8 7 6 5 4 2 t + 4 t - 4 t + t - 4 t + 4 t - 2 t - t + 2 t - 1 the first , 30, terms are , [2, 4, 8, 15, 28, 52, 92, 161, 278, 476, 808, 1368, 2300, 3852, 6424, 10687, 17724, 29348, 48496, 80027, 131860, 217052, 356892, 586396, 962742, 1579740, 2590664, 4246725, 6958460, 11398128] the asympt. constant is, 1.634408012 For words avoiding the pattern, [1, 0, 1, 0], with spacings <=, 1 2 7 9 5 6 10 12 the generating function is, (-t + 2 t + 2 t - 2 t - 3 t + 2 t + t 11 8 4 14 / 6 5 4 2 3 - 2 t + 3 t - 3 t + t - 1) / (t + 4 t - 4 t - t + 2 t - 1 / 13 14 12 8 9 10 - 2 t + 2 t + t + 4 t - 3 t - 6 t + 3 t ) the first , 30, terms are , [2, 4, 8, 15, 28, 53, 92, 157, 270, 459, 794, 1384, 2388, 4111, 7048, 12044, 20654, 35497, 61032, 104960, 180262, 309244, 530542, 910469, 1563204, 2684824, 4610694, 7916118, 13588918, 23325395] the asympt. constant is, 1.716801340 The low champion is, [1, 1, 0, 0], with asympt. constant, 1.634408012 The high champion is, [0, 0, 0, 0], with asympt. constant, 1.886514351 For words avoiding the pattern, [0, 1, 1, 0], with spacings <=, 2 58 57 30 28 17 the generating function is, - (-1 - 2 t - 2 t - 28 t + 7 t + 3 t 18 19 3 63 27 8 9 29 22 + 55 t - 24 t - t + t - 38 t - 6 t - 12 t - 8 t - 5 t 13 11 12 14 15 10 25 23 + 5 t + 2 t + 14 t + 3 t + 6 t + 15 t + 35 t - 25 t 24 26 5 6 7 21 62 60 20 16 + 22 t + 16 t - 2 t - 4 t - 2 t + t + t + t + 8 t - 2 t 61 59 55 56 54 53 48 46 49 51 + t - t - 3 t - 3 t + 6 t + t - 4 t - 4 t - 4 t + 5 t 38 33 40 41 42 50 31 32 45 + 15 t - 23 t + 10 t + t - 8 t - 6 t + t + 28 t + 10 t 36 34 35 37 43 44 52 47 - 17 t - 20 t - 16 t + 17 t - 4 t - 13 t + 6 t - 8 t 39 / 57 30 28 17 18 19 + 26 t ) / (1 - 2 t - t + 8 t - 55 t + 6 t - 36 t + 52 t / 3 64 4 27 8 9 29 22 13 11 + t + t - t + 49 t + t + 2 t + 22 t - 7 t + 5 t + 9 t 12 14 15 10 25 23 24 26 - 8 t + 3 t - 7 t - 10 t - 15 t + 22 t - 25 t + 13 t 5 7 21 60 20 16 59 55 54 + 2 t - t + 7 t - 2 t - 20 t + 5 t - t + 10 t - 5 t 53 48 46 51 38 33 40 41 + 5 t - 5 t + 23 t - 13 t + 5 t + 46 t + 18 t + 6 t 42 50 31 32 45 36 34 35 37 - 2 t + t - 16 t - 26 t - 25 t - 8 t - 5 t - t - 15 t 43 44 52 47 39 - 4 t + 11 t - t + 4 t - 17 t ) the first , 30, terms are , [2, 4, 8, 15, 28, 52, 92, 162, 282, 463, 760, 1229, 1997, 3211, 5177, 8349, 13469, 21739, 35266, 57203, 93043, 151269, 246440, 401313, 653908, 1065296, 1735877, 2827045, 4604510, 7496515] the asympt. constant is, 1.627709813 For words avoiding the pattern, [0, 0, 0, 0], with spacings <=, 2 30 28 17 18 19 the generating function is, - (1 + t - 13 t - 14 t - 3 t + 15 t + t 3 2 4 27 8 9 29 22 13 + 4 t + 2 t + 5 t + t + 18 t + 36 t - 15 t + 2 t - 10 t 11 12 14 15 10 25 23 24 + 23 t - 3 t - 43 t - 50 t + 29 t + 19 t + 2 t + 26 t 26 5 6 7 21 20 16 33 31 + 17 t + 7 t + 14 t + 14 t - 16 t + 3 t - 31 t + 2 t - 2 t 32 36 34 37 / 30 28 17 18 - 2 t - t - 2 t + t ) / (-1 + t - 5 t + 6 t - 26 t - 9 t / 19 4 27 8 9 22 13 11 12 + 8 t + 2 t + 11 t + 7 t - t - 9 t - 5 t + 11 t + 19 t 14 15 10 25 23 24 26 5 6 + 9 t - 6 t + 9 t + 12 t + 2 t - 12 t + 2 t + 3 t - t 7 21 20 16 38 33 31 32 36 34 + 7 t + 10 t + t - 18 t + t - 3 t - 9 t - t + t + 2 t 35 37 - t - 2 t ) the first , 30, terms are , [2, 4, 8, 15, 29, 56, 103, 192, 364, 668, 1222, 2233, 4153, 7728, 14341, 26637, 49454, 91751, 169960, 314994, 584343, 1083605, 2008967, 3724431, 6906845, 12808759, 23751299, 44043289, 81672882, 151449540] the asympt. constant is, 1.854320341 For words avoiding the pattern, [1, 0, 0, 0], with spacings <=, 2 17 18 19 3 2 4 the generating function is, - (1 + t - 3 t - 3 t + 3 t + t + t + t 8 9 22 13 11 12 14 15 10 + 6 t + 6 t + 9 t - 12 t - 7 t - 11 t - 10 t - 7 t - t 25 23 24 5 6 7 21 20 16 / + t + 7 t + 3 t + 2 t + 2 t + 4 t + 8 t + 6 t - 4 t ) / ( / 28 17 18 19 3 2 27 8 9 -1 + t + 2 t + 2 t - 5 t + 2 t + t + t + 4 t - t + 4 t 22 13 11 12 14 15 10 23 24 - 5 t + 4 t - 9 t - t + 10 t + 7 t - 8 t - 8 t - 6 t 26 5 6 7 21 20 16 + 4 t - t + t - 3 t - 4 t + 2 t + 7 t ) the first , 30, terms are , [2, 4, 8, 15, 28, 52, 94, 169, 304, 525, 905, 1563, 2684, 4607, 7909, 13521, 23109, 39486, 67304, 114702, 195540, 333115, 567456, 966630, 1645891, 2802381, 4771487, 8122564, 13827058, 23538333] the asympt. constant is, 1.702057535 For words avoiding the pattern, [0, 1, 0, 0], with spacings <=, 2 28 33 34 29 17 the generating function is, - (1 + t - 15 t - 6 t - 2 t - 19 t - 20 t 18 19 3 30 31 2 4 27 8 - 26 t - 27 t + 2 t - 16 t - 14 t + t + 2 t - 17 t + 5 t 32 9 36 22 13 11 12 14 - 12 t + 18 t + t + 4 t - 31 t - 11 t - 21 t - 27 t 15 10 25 23 24 26 5 6 7 - 23 t + 6 t - 14 t + 4 t - 11 t - 13 t + 3 t + 7 t + 5 t 21 20 16 / 28 33 34 35 - 10 t - 17 t - 27 t ) / (-1 + t - 3 t - 6 t - 4 t - 2 t / 29 17 18 19 30 31 2 4 27 8 + 3 t - 8 t + t - 4 t - 5 t - 2 t + t + t + t + 2 t 32 9 36 22 13 11 12 14 15 - 2 t - 10 t - t - 16 t + 5 t + 7 t - 2 t - 7 t - 7 t 10 25 23 24 26 6 7 37 21 + 9 t - 2 t - 2 t + 13 t - 6 t - 2 t + 2 t + t - 4 t 20 16 - 10 t + 5 t ) the first , 30, terms are , [2, 4, 8, 15, 28, 52, 91, 161, 284, 486, 825, 1401, 2408, 4138, 7104, 12261, 21153, 36472, 62944, 108684, 187696, 323761, 558470, 963559, 1661940, 2866549, 4944475, 8528908, 14711499, 25374654] the asympt. constant is, 1.724962815 For words avoiding the pattern, [1, 1, 0, 0], with spacings <=, 2 36 34 32 26 25 24 23 the generating function is, (t - t + t - 3 t + 2 t + 4 t - 2 t 22 20 18 17 14 12 11 10 8 7 - 2 t - 7 t + t + 4 t - 4 t - 12 t + 2 t + 9 t - t - 4 t 5 / 28 33 34 35 29 18 3 - 2 t - 1) / ((1 - t - 6 t + t + t + t - 6 t + 10 t - t / 30 31 2 27 8 32 9 36 22 13 39 + 2 t + 2 t - t + 3 t - t - t - t - t + 3 t - 15 t + t 11 12 14 15 10 25 23 24 26 + 6 t - 3 t + t + 5 t + 2 t + 8 t + 3 t - 2 t + 7 t 5 6 7 37 21 20 38 + 2 t - 2 t + 6 t - t - 4 t - 18 t - t ) (-1 + t)) the first , 30, terms are , [2, 4, 8, 15, 28, 52, 92, 161, 278, 460, 760, 1265, 2068, 3363, 5454, 8762, 14042, 22475, 35738, 56766, 90110, 142544, 225060, 355144, 559296, 880067, 1383952, 2173698, 3411640, 5353127] the asympt. constant is, 1.562013941 For words avoiding the pattern, [1, 0, 1, 0], with spacings <=, 2 34 17 18 19 30 2 the generating function is, (1 + 4 t - 8 t - 21 t - 6 t + 4 t + t 4 8 32 9 22 13 11 12 14 15 + 3 t + 7 t - 8 t + 4 t + 2 t - 8 t - 4 t - 6 t - t - 2 t 10 25 23 26 5 6 7 21 20 16 - 4 t + 8 t + 4 t + 3 t + 2 t + 9 t + 4 t + 2 t - 8 t + t / 28 29 17 18 19 3 30 2 ) / (1 - 2 t - t - 10 t - 10 t + 7 t - 4 t - 2 t - 2 t + t / 4 8 32 9 22 13 11 12 14 + 4 t + t + 3 t - 8 t - 9 t - 6 t - 2 t + 9 t + 8 t 15 10 25 23 24 26 5 6 7 - 10 t + 13 t + 6 t - 2 t + 19 t + 7 t - 4 t + 5 t - 6 t 21 20 16 - 16 t + 4 t + 3 t ) the first , 30, terms are , [2, 4, 8, 15, 28, 53, 92, 157, 270, 435, 704, 1142, 1832, 2996, 4902, 8010, 13146, 21443, 34824, 56450, 91248, 147534, 238918, 387322, 628922, 1022198, 1661054, 2697971, 4378950, 7101989] the asympt. constant is, 1.622474524 The low champion is, [1, 1, 0, 0], with asympt. constant, 1.562013941 The high champion is, [0, 0, 0, 0], with asympt. constant, 1.854320341 For words avoiding the pattern, [0, 1, 1, 0], with spacings <=, 3 110 82 92 80 the generating function is, - (-1 - 263 t - 1944 t + 758 t + 1115 t 130 77 76 88 17 18 19 + 4 t + 1579 t - 924 t - 986 t - 32 t + 18 t - 22 t 93 67 3 24 127 8 9 119 - 21 t - 1013 t - t + 199 t - 8 t - 6 t - 15 t + 35 t 120 106 117 99 97 22 113 + 71 t - 133 t + 20 t - 161 t + 286 t - 33 t - 95 t 74 73 72 115 96 81 83 - 70 t - 2569 t + 1301 t + 28 t + 169 t + 1251 t - 11 t 112 78 111 13 125 27 11 - 40 t + 4119 t + 97 t + 19 t - 22 t - 106 t - 5 t 12 123 107 58 129 14 128 - 11 t + 42 t - 270 t + 1281 t + 4 t + 37 t - 12 t 36 108 102 114 109 15 10 - 329 t + 358 t + 174 t + 85 t - 32 t + 3 t + 7 t 122 25 23 26 39 79 60 + 12 t - 20 t + 101 t - 39 t - 1172 t - 257 t + 1621 t 63 68 56 94 116 55 57 + 643 t - 3101 t + 181 t + 922 t + 36 t + 117 t + 1027 t 98 101 6 21 29 28 100 + 315 t - 100 t - 4 t + 70 t + 86 t - 286 t - 414 t 66 124 20 16 34 35 33 - 509 t - 26 t + 57 t - 16 t + 30 t + 551 t + 143 t 103 37 38 91 87 71 65 - 320 t - 47 t + 297 t - 193 t - 990 t - 2455 t - 2257 t 43 95 5 121 105 84 85 + 375 t + 1111 t - 2 t - 26 t + 530 t - 624 t - 2301 t 31 30 32 86 90 118 64 - 195 t + 86 t + 189 t - 261 t - 767 t - 63 t + 939 t 61 7 69 54 49 48 46 + 2895 t - 2 t + 604 t - 207 t + 7 t + 390 t + 1739 t 45 47 104 51 44 50 62 - 586 t + 700 t - 82 t - 118 t + 30 t - 1073 t - 320 t 126 75 40 42 41 52 53 + 8 t + 3911 t - 225 t - 1319 t - 153 t - 97 t - 1313 t 89 70 59 / 110 82 92 + 495 t - 2451 t + 536 t ) / (1 + 193 t + 3062 t - 1403 t / 80 77 76 88 17 18 19 - 2 t - 2604 t - 3661 t + 4273 t + 222 t + 2 t - 22 t + 36 t 93 67 3 24 127 4 8 9 119 + 590 t + 1637 t + t - 19 t + 16 t - t + t + 5 t - 94 t 120 106 99 97 22 113 74 - 36 t + 754 t + 204 t + 371 t + 53 t + 51 t - 3487 t 73 72 115 96 81 83 112 + 4132 t - 2427 t + 73 t + 562 t + 424 t - 2280 t + 205 t 78 111 13 125 27 12 123 - 1440 t - 390 t - 11 t - 4 t + 27 t + 3 t - 30 t 107 58 129 14 128 36 108 + 39 t + 1378 t - 16 t - 6 t + 4 t + 503 t - 664 t 102 114 109 15 10 122 25 - 462 t - 210 t + 487 t + 16 t - 8 t - 38 t + 69 t 23 26 39 79 60 63 68 - 75 t + 2 t + 729 t + 3911 t - 1343 t - 1414 t + 770 t 56 94 116 55 57 98 101 - 103 t - 286 t - 4 t + 723 t - 528 t - 38 t - 347 t 21 29 28 100 66 124 20 - 11 t - 57 t - 35 t + 488 t - 1708 t + 68 t - 25 t 16 34 35 33 103 37 38 + 5 t - 226 t - 29 t + 151 t + 603 t - 616 t - 9 t 91 87 71 65 43 95 5 - 224 t + 1057 t - 774 t + 1063 t - 1374 t - 419 t + 2 t 121 105 84 85 31 30 32 + 97 t - 608 t + 1161 t + 1852 t - 46 t + 121 t - 27 t 86 90 118 64 61 7 69 - 2062 t + 1227 t + 83 t + 736 t - 521 t - t - 2440 t 54 49 48 46 45 47 104 - 711 t - 229 t - 99 t - 1387 t + 709 t + 750 t - 270 t 51 44 50 62 126 131 75 - 880 t + 522 t + 510 t + 911 t - 30 t + 4 t - 2763 t 40 42 41 52 53 89 70 - 1038 t + 1026 t + 405 t + 54 t + 603 t - 1933 t + 4064 t 59 + 226 t ) the first , 30, terms are , [2, 4, 8, 15, 28, 52, 92, 162, 282, 463, 760, 1229, 1952, 3042, 4736, 7351, 11346, 17371, 26659, 40761, 62329, 95485, 146684, 225435, 347313, 536456, 830618, 1287061, 1998403, 3107095] the asympt. constant is, 1.553640492 For words avoiding the pattern, [0, 0, 0, 0], with spacings <=, 3 110 82 92 the generating function is, - (1 + 2527 t + 9128 t + 18981 t + t 80 130 77 76 88 17 - 17630 t + 13 t + 491 t + 22175 t - 11189 t - 329 t 18 19 93 67 3 24 2 - 529 t - 820 t + 14983 t - 2112 t + 4 t + 931 t + 2 t 127 4 8 9 119 120 106 - 128 t + 7 t + 42 t + 68 t + 550 t + 473 t - 5319 t 117 99 97 22 113 74 - 1311 t - 7443 t - 14652 t - 472 t + 1388 t + 15837 t 73 72 115 96 81 83 + 993 t - 17251 t - 1732 t - 16058 t - 5652 t + 11133 t 112 78 111 13 125 27 11 + 2235 t - 13184 t + 1768 t + 158 t - 43 t + 93 t + 126 t 12 123 107 58 129 14 128 + 197 t + 62 t - 4165 t + 15824 t + 17 t + 107 t + 3 t 36 108 102 114 109 15 + 5293 t - 2115 t + 10185 t - 1001 t + 947 t + 50 t 10 122 25 23 26 39 79 + 89 t + 417 t + 1524 t + 352 t + 1406 t - 7359 t - 13663 t 60 63 68 56 94 116 - 1510 t - 14006 t + 4836 t + 5499 t + 7199 t - 1297 t 55 57 98 101 6 21 + 2014 t + 20173 t - 15844 t + 9064 t + 18 t - 872 t 29 28 100 66 124 20 16 - 2867 t - 971 t + 4404 t - 3604 t + 67 t - 674 t - 308 t 34 35 33 103 37 38 + 4716 t + 5752 t + 76 t + 8140 t + 2168 t - 5188 t 91 87 71 65 43 95 + 21077 t - 11648 t - 19580 t - 11497 t + 12549 t - 9940 t 5 121 105 84 85 31 + 9 t + 502 t - 4145 t + 17538 t + 18474 t - 3935 t 30 32 86 90 118 64 - 4097 t - 2771 t - 1379 t + 1414 t - 179 t - 18122 t 61 7 69 54 49 48 - 19617 t + 22 t + 1260 t + 1819 t - 16069 t - 9305 t 46 45 47 104 51 44 + 15510 t + 22720 t + 3805 t + 1654 t - 6156 t + 20321 t 50 62 126 131 75 40 - 16016 t - 22091 t - 203 t + 3 t + 27060 t - 7737 t 42 41 52 53 89 70 + 5223 t - 4751 t - 392 t - 227 t - 13798 t - 12690 t 59 / 110 82 92 80 130 + 6262 t ) / (-1 - 87 t - 3562 t + 309 t + t - 2807 t + 4 t / 77 76 88 17 18 19 93 + 12659 t + 3303 t - 645 t - 72 t - 4 t - 17 t + 1535 t 67 24 127 8 9 119 120 + 660 t - 154 t - 65 t - 3 t + 15 t - 439 t - 56 t 106 117 99 97 22 113 74 + 986 t + 81 t - 6210 t - 2858 t - 289 t + 934 t - 5987 t 73 72 115 96 81 83 - 5647 t - 838 t + 295 t - 2553 t - 9963 t + 1870 t 112 78 111 13 125 27 11 - 956 t + 8649 t + 1068 t + 55 t + 96 t + 661 t + 14 t 12 123 107 58 129 14 128 + 5 t + 290 t - 32 t + 3192 t - 14 t + 21 t - 128 t 36 108 102 114 109 15 10 + 643 t - 3038 t + 7 t + 1507 t - 2201 t + 33 t + 11 t 122 25 23 26 39 79 60 + 249 t - 118 t - 376 t + 129 t + 534 t + 2610 t + 4164 t 63 68 56 94 116 55 - 8484 t - 7214 t + 1015 t + 8262 t - 491 t + 2720 t 57 98 101 6 21 29 28 - 4700 t + 1043 t - 873 t - t + 6 t + 512 t + 434 t 100 66 124 20 16 34 35 - 8513 t - 3289 t - 133 t - 271 t + 153 t - 1730 t - 321 t 33 103 37 38 91 87 - 1362 t + 2136 t + 1808 t + 3202 t - 8122 t + 2697 t 71 65 43 95 5 121 105 + 106 t - 8477 t - 1125 t + 11319 t + 5 t + 67 t + 3604 t 84 85 31 30 32 86 - 7033 t + 3014 t - 503 t - 207 t - 1176 t + 12626 t 90 118 64 61 7 69 54 - 7318 t - 883 t - 2470 t + 10887 t + 7 t - 2861 t - 14 t 49 48 46 45 47 104 + 3139 t + 8278 t + 6291 t + 1318 t + 5285 t + 3616 t 51 44 50 62 126 131 75 - 6788 t - 835 t - 984 t - 1364 t + 164 t + 10 t - 7412 t 40 42 41 52 53 89 - 142 t - 3717 t - 2151 t - 4775 t - 3972 t + 4279 t 70 59 132 + 7077 t + 3407 t + 3 t ) the first , 30, terms are , [2, 4, 8, 15, 29, 56, 103, 192, 364, 668, 1222, 2233, 3987, 7138, 12903, 23394, 42893, 78548, 143718, 262561, 479162, 875858, 1598383, 2916885, 5320377, 9699388, 17690229, 32265491, 58852319, 107350924] the asympt. constant is, 1.824372287 For words avoiding the pattern, [1, 0, 0, 0], with spacings <=, 3 17 18 19 67 3 the generating function is, (1 + t - 244 t - 285 t - 284 t + 6 t + 2 t 24 2 4 8 9 22 73 72 13 + 250 t + 2 t + 3 t + 11 t + 16 t - 37 t + t + t - 16 t 27 11 12 58 14 36 15 + 717 t + 18 t + 13 t - 235 t - 56 t - 590 t - 104 t 10 25 23 26 39 60 63 + 15 t + 413 t + 85 t + 617 t - 1445 t - 222 t - 73 t 68 56 55 57 6 21 29 28 + 16 t - 65 t + 150 t - 175 t + 5 t - 184 t + 885 t + 822 t 66 20 16 34 35 33 37 + 10 t - 250 t - 173 t + 130 t - 253 t + 386 t - 943 t 38 71 65 43 5 31 30 - 1208 t + 5 t - 11 t - 1083 t + 4 t + 791 t + 835 t 32 64 61 7 69 54 49 48 + 649 t - 33 t - 174 t + 8 t + 8 t + 348 t + 1062 t + 868 t 46 45 47 51 44 50 62 + 161 t - 233 t + 587 t + 1014 t - 687 t + 1103 t - 125 t 40 42 41 52 53 70 59 - 1556 t - 1341 t - 1514 t + 852 t + 616 t + 7 t - 255 t ) / 17 18 19 67 3 24 22 / (1 - t + 9 t - 28 t - 68 t + 2 t - 2 t + 17 t - 53 t / 13 27 11 12 58 14 36 15 + 20 t + 105 t + 8 t + t - 18 t - 3 t - 221 t + 20 t 10 25 23 26 39 60 63 68 + 9 t + 68 t - 2 t + 78 t - 149 t + 15 t + 23 t + 2 t 56 55 57 6 21 29 28 66 - 78 t - 97 t - 36 t - t - 45 t + 119 t + 196 t + 8 t 20 16 34 35 33 37 38 - 62 t - 31 t - 108 t - 203 t - 69 t - 257 t - 217 t 65 43 5 31 30 32 64 61 + 10 t + 144 t - t + 81 t + 121 t + 21 t + 11 t + 21 t 7 54 49 48 46 45 47 51 + 2 t - 88 t + 63 t + 99 t + 196 t + 208 t + 198 t - 53 t 44 50 62 40 42 41 52 53 + 209 t + 14 t + 16 t - 91 t + 74 t - 2 t - 89 t - 107 t 59 - 9 t ) the first , 30, terms are , [2, 4, 8, 15, 28, 52, 94, 169, 304, 525, 905, 1563, 2630, 4429, 7468, 12513, 20952, 35120, 58693, 98077, 163867, 273175, 455447, 759513, 1265299, 2108243, 3513131, 5852240, 9748627, 16239791] the asympt. constant is, 1.665364762 For words avoiding the pattern, [0, 1, 0, 0], with spacings <=, 3 124 98 17 18 19 the generating function is, - (1 + 48 t + 291 t - 7 t - 47 t + 103 t 67 3 24 9 22 73 72 13 + 1331 t + t - 534 t + 13 t + 155 t - 574 t + 466 t - 95 t 27 120 128 11 12 58 14 + 15 t + 56 t + 4 t - 9 t + 32 t + 1218 t - 117 t 36 15 10 93 104 103 107 + 771 t + 21 t - 6 t + 1995 t - 74 t + 167 t - 188 t 25 23 26 39 78 60 63 + 479 t - 55 t + 438 t - 707 t - 3189 t + 32 t + 2315 t 68 56 82 106 108 109 55 - 2479 t - 3542 t - 3253 t - 319 t - 51 t - 25 t - 1899 t 57 88 87 86 129 6 117 21 + 541 t + 52 t - 179 t + 489 t + 8 t + 4 t - 68 t + 27 t 90 89 91 92 118 119 29 + 2023 t + 407 t + 2341 t + 2669 t + 21 t + 62 t - 101 t 28 66 20 16 34 35 33 37 + 138 t - 61 t + 275 t - t - 318 t + 206 t - 519 t - 233 t 38 101 110 84 71 65 111 - 1052 t - 321 t - 146 t - 2925 t + 45 t + 2860 t - 288 t 79 43 83 81 105 5 31 - 3411 t - 1787 t - 3577 t - 1775 t - 253 t + t + 770 t 30 32 112 114 113 115 116 - 4 t - 623 t - 244 t - 50 t - 96 t - 92 t - 87 t 64 61 7 130 69 95 54 + 2634 t + 1113 t - 2 t + 4 t + 626 t + 1446 t - 81 t 49 48 46 80 96 45 47 - 2096 t + 2063 t - 1443 t - 1854 t + 962 t - 410 t + 339 t 51 44 50 85 74 77 75 - 1809 t - 105 t + 981 t - 2891 t + 768 t - 1027 t + 1408 t 97 62 102 125 127 40 42 + 1295 t + 2003 t - 235 t + 26 t + 12 t - 1196 t + 435 t 41 76 52 53 70 59 121 - 579 t + 875 t + 176 t + 3477 t + 1742 t + 2732 t + 40 t 94 126 123 122 100 99 / + 1687 t + 20 t + 42 t + 30 t - 71 t + 281 t ) / (-1 / 124 98 17 18 19 67 3 + 2 t - 6 t + 720 t - 6 t + 21 t - 14 t - 599 t - t 24 4 9 22 73 72 13 27 + 260 t + t - 12 t - 19 t + 1150 t + 256 t + 67 t + 392 t 120 128 11 12 58 14 36 15 + 6 t + 8 t + 2 t - 35 t - 57 t + 8 t - 168 t - 86 t 10 93 104 103 107 25 23 + 13 t + 672 t + 138 t - 520 t - 160 t - 551 t + 69 t 26 39 78 60 63 68 56 + 51 t - 716 t + 1206 t + 2231 t + 343 t + 2981 t + 999 t 82 106 108 109 55 57 88 + 996 t + 145 t - 179 t - 101 t + 1963 t - 2271 t + 5 t 87 86 129 6 117 21 90 + 652 t - 2580 t - 4 t - 2 t - 5 t + 124 t - 908 t 89 91 92 118 119 29 28 - 157 t + 97 t - 180 t - 85 t - 33 t + 149 t - 263 t 66 20 16 34 35 33 37 + 2678 t - 120 t + 33 t - 561 t + 38 t + 230 t - 156 t 38 101 110 84 71 65 111 + 816 t + 403 t + 100 t - 305 t + 1541 t - 2035 t + 160 t 79 43 83 81 105 5 31 30 - 603 t + 1463 t - 108 t - 2 t + 180 t - t - 847 t + 311 t 32 112 114 113 115 116 64 + 794 t - 36 t - 54 t - 152 t + 46 t + 13 t + 479 t 61 7 131 130 69 95 54 + 21 t + 4 t + 4 t + 4 t - 1397 t + 400 t + 1540 t 49 48 46 80 96 45 47 + 3972 t - 2488 t + 1958 t - 1248 t + 670 t - 210 t - 2080 t 51 44 50 85 74 77 75 + 1934 t - 722 t - 2500 t + 373 t - 1505 t + 997 t - 1064 t 97 62 102 125 127 40 42 - 402 t - 801 t - 37 t + 22 t + 8 t + 258 t - 1704 t 41 76 52 53 70 59 121 + 515 t - 78 t - 159 t - 3018 t - 1944 t - 1709 t + 20 t 94 126 123 122 100 99 - 66 t + 6 t - 12 t + 14 t + 369 t - 142 t ) the first , 30, terms are , [2, 4, 8, 15, 28, 52, 91, 161, 284, 486, 825, 1401, 2299, 3777, 6277, 10538, 17644, 29436, 49626, 83625, 140319, 236246, 397980, 669870, 1128246, 1899976, 3198599, 5377077, 9042226, 15219062] the asympt. constant is, 1.682108105 For words avoiding the pattern, [1, 1, 0, 0], with spacings <=, 3 17 18 19 24 9 the generating function is, - (1 - 46 t - 34 t - 16 t - 113 t + 6 t 22 13 27 11 12 58 14 36 - 92 t - 26 t - 244 t + 2 t + 12 t + 3 t + 10 t + 196 t 15 10 25 23 26 39 60 56 - 12 t - 5 t + 30 t + 44 t - 142 t - 146 t + 6 t - 69 t 55 57 21 29 28 20 16 - 12 t - 22 t + 78 t + 156 t + 131 t + 111 t + 20 t 34 35 33 37 38 43 5 31 - 199 t - 18 t - 280 t + 550 t + 108 t + 120 t + 2 t - 80 t 30 32 61 7 54 49 48 46 + 205 t + 6 t + 6 t + 4 t + 9 t - 102 t - 89 t + 97 t 45 47 51 44 50 62 40 + 154 t - 106 t + 134 t + 60 t - 37 t + 4 t - 190 t 42 41 52 53 59 8 / 17 - 148 t - 176 t + 82 t + 16 t + 28 t + t ) / (-1 + 2 t + 2 t / 18 19 24 4 9 22 13 27 - 52 t + 70 t - 136 t - t - 6 t - 98 t - 18 t + 130 t 12 58 14 36 15 10 25 23 + 17 t + 102 t + 4 t + 509 t - 16 t + 5 t + 98 t + 92 t 26 39 60 63 56 55 57 6 - 142 t - 468 t + 20 t - 10 t + 49 t + 74 t - 206 t + 4 t 21 29 28 66 20 16 34 35 + 98 t + 118 t - 74 t + t - 54 t + 21 t + 220 t - 322 t 33 37 38 43 5 31 30 32 - 180 t - 426 t + 579 t - 176 t - 2 t - 166 t + 69 t + 53 t 64 61 7 54 49 48 46 45 + 3 t - 40 t - 8 t + 21 t - 114 t + 125 t + 197 t - 554 t 47 51 44 50 62 40 42 + 14 t - 60 t + 403 t - 17 t + 46 t + 306 t + 295 t 41 52 53 59 8 - 468 t + 332 t - 230 t - 40 t + 7 t ) the first , 30, terms are , [2, 4, 8, 15, 28, 52, 92, 161, 278, 460, 760, 1265, 2018, 3245, 5150, 8102, 12628, 19626, 30140, 46470, 70998, 107960, 163600, 247524, 371826, 559175, 835762, 1248882, 1859290, 2764964] the asympt. constant is, 1.434009501 For words avoiding the pattern, [1, 0, 1, 0], with spacings <=, 3 68 79 81 the generating function is, (1 - 56202 t + 2 t - 6242 t - 4972 t 17 18 19 3 24 2 4 9 + 414 t + 316 t + 178 t + 6 t - 3008 t + 4 t + 11 t + 138 t 70 85 22 87 13 27 92 - 44910 t - 66 t - 745 t + 298 t + 478 t - 11220 t + 6 t 88 91 11 12 58 14 36 + 197 t + 22 t + 286 t + 399 t - 74338 t + 542 t - 23603 t 15 10 25 23 26 39 + 548 t + 201 t - 5154 t - 1508 t - 8079 t + 10190 t 60 63 56 55 57 86 - 78102 t - 74316 t - 62176 t - 51266 t - 69858 t + 249 t 89 90 6 21 29 28 66 + 106 t + 51 t + 34 t - 316 t - 18240 t - 14664 t - 63982 t 20 84 16 34 35 33 - 49 t - 864 t + 495 t - 30286 t - 28462 t - 30230 t 71 74 77 37 38 93 - 37440 t - 14188 t - 5966 t - 15584 t - 3779 t + 4 t 82 75 43 5 31 30 - 3601 t - 9304 t + 74566 t + 18 t - 25738 t - 22107 t 32 67 64 61 7 69 - 28571 t - 60356 t - 71111 t - 78054 t + 54 t - 51152 t 65 54 49 48 46 45 - 67550 t - 36786 t + 64824 t + 80217 t + 94992 t + 93760 t 47 51 44 50 62 40 + 90526 t + 23014 t + 86551 t + 44780 t - 76681 t + 26042 t 42 41 52 53 80 59 + 59224 t + 42456 t + 968 t - 19138 t - 5976 t - 76910 t 83 8 78 73 72 76 / - 2116 t + 89 t - 6089 t - 21122 t - 29199 t - 6728 t ) / (1 / 68 79 81 17 18 19 3 24 - 1313 t - 270 t - 134 t + 46 t + 38 t + 82 t - 2 t + 69 t 9 70 85 22 87 13 27 88 - 4 t - 1312 t - 74 t + 61 t - 36 t - 22 t - 188 t - 7 t 91 11 12 58 14 36 15 10 + 4 t - 10 t - 20 t + 257 t - 10 t - 189 t + 18 t - t 25 23 26 39 60 63 56 + 84 t + 98 t - 26 t + 694 t - 493 t - 1200 t - 87 t 55 57 86 89 90 6 21 29 - 702 t + 172 t - 65 t + 10 t + 6 t + t + 100 t - 532 t 28 66 20 84 16 34 35 - 345 t - 995 t + 38 t - 87 t + 17 t - 392 t - 382 t 33 71 74 77 37 38 82 - 576 t - 1124 t - 528 t - 442 t - 122 t + 261 t - 83 t 75 43 5 31 30 32 67 - 468 t + 1586 t - 2 t - 732 t - 617 t - 540 t - 1136 t 64 61 7 69 65 54 49 - 1084 t - 1012 t - 2 t - 1322 t - 1036 t - 1125 t + 136 t 48 46 45 47 51 44 50 + 220 t + 536 t + 896 t + 440 t - 742 t + 1105 t - 316 t 62 40 42 41 52 53 - 1161 t + 1146 t + 1628 t + 1474 t - 1138 t - 1218 t 80 59 83 8 78 73 72 76 - 186 t - 64 t - 74 t + 4 t - 376 t - 630 t - 864 t - 464 t ) the first , 30, terms are , [2, 4, 8, 15, 28, 53, 92, 157, 270, 435, 704, 1142, 1734, 2646, 4018, 6012, 9144, 13987, 21346, 32693, 50488, 77519, 118702, 181699, 275542, 416680, 629564, 948711, 1430378, 2161496] the asympt. constant is, 1.518286691 The low champion is, [1, 1, 0, 0], with asympt. constant, 1.434009501 The high champion is, [0, 0, 0, 0], with asympt. constant, 1.824372287 This took, 295.408, seconds