The first , 30, terms of the counting sequence for binary words avoiding the set of patterns, {[1, 1, 1, 1], [0, 0, 0, 0]}, with spacings <= , 0, are [2, 4, 8, 14, 26, 48, 88, 162, 298, 548, 1008, 1854, 3410, 6272, 11536, 21218, 39026, 71780, 132024, 242830, 446634, 821488, 1510952, 2779074, 5111514, 9401540, 17292128, 31805182, 58498850, 107596160] 3 2 t + t + t + 1 The generating function is, - --------------- 3 2 t + t + t - 1 n the asymptotics seems to be roughly, 1.2351 1.8393 The first , 30, terms of the counting sequence for binary words avoiding the set of patterns, {[1, 1, 1, 1], [0, 0, 0, 0]}, with spacings <= , 1, are [2, 4, 8, 14, 26, 48, 78, 132, 230, 398, 690, 1190, 2046, 3528, 6090, 10510, 18132, 31274, 53942, 93052, 160528, 276924, 477702, 824050, 1421516, 2452184, 4230144, 7297190, 12587970, 21714804] 13 10 8 7 5 4 3 The generating function is, - (3 t - 3 t + t - t - 4 t - 3 t - 2 t - t 14 11 9 6 2 / - 1 + 2 t - 2 t + 2 t - 6 t - 2 t ) / ( / 11 10 6 5 4 3 2 2 (t - t + t - 2 t + t - t + t - 2 t + 1) (t + t + 1)) n the asymptotics seems to be roughly, 1.7180 1.7250 The first , 30, terms of the counting sequence for binary words avoiding the set of patterns, {[1, 1, 1, 1], [0, 0, 0, 0]}, with spacings <= , 2, are [2, 4, 8, 14, 26, 48, 78, 132, 230, 356, 548, 842, 1338, 2146, 3470, 5634, 9074, 14522, 23060, 36684, 58582, 93882, 150540, 241254, 386384, 618498, 989668, 1583196, 2532846, 4053508] 39 23 22 26 The generating function is, - (1 + 73 t + 31 t - 161 t + 2 t - 40 t 25 34 28 2 3 20 19 37 + 92 t + 41 t + 107 t + 3 t + 5 t - 354 t - 263 t + 22 t 10 42 45 43 50 49 46 6 + 53 t + 21 t + 23 t + 10 t + 8 t + 8 t + 16 t + 21 t 7 8 9 11 13 48 12 41 4 + 27 t + 35 t + 55 t + 17 t - 70 t + 6 t - 45 t + 22 t + 8 t 5 35 36 14 40 15 16 31 + 13 t + 51 t + 11 t - 102 t + 45 t - 154 t - 116 t - 18 t 30 33 27 44 24 32 29 38 - 34 t - 39 t + 82 t + 20 t + 188 t - 49 t + 66 t + 89 t 17 18 21 / 39 23 22 26 - 50 t - 89 t - 327 t ) / (-1 + t + 15 t + 11 t - 10 t / 25 34 28 2 3 20 19 10 42 45 - 10 t + 5 t - t + t + t - 2 t - 9 t + 3 t - t + t 6 7 8 9 11 13 12 41 5 35 36 + 3 t + t + t + 5 t - 5 t - 4 t - 3 t + 2 t + t + 5 t - t 14 40 15 16 31 33 27 24 32 - 6 t + t - 8 t - 2 t - 2 t + 3 t + 2 t + 12 t - 11 t 29 38 18 21 + 8 t - t - 13 t - 9 t ) n the asymptotics seems to be roughly, 3.0159 1.6006 The first , 30, terms of the counting sequence for binary words avoiding the set of patterns, {[1, 1, 1, 1], [0, 0, 0, 0]}, with spacings <= , 3, are [2, 4, 8, 14, 26, 48, 78, 132, 230, 356, 548, 842, 1078, 1344, 1764, 2356, 3388, 4804, 6784, 9622, 13674, 19732, 27952, 39966, 55762, 77642, 109188, 153436, 217026, 306882] 39 23 22 123 The generating function is, - (1 - 9224 t + 304 t + 24 t + 1310 t 125 74 26 25 34 28 + 1228 t + 2 t + 3714 t - 76 t + 612 t + 8529 t - 3252 t 2 3 20 129 19 37 104 + 4 t + 7 t - 741 t + 1170 t - 970 t - 696 t - 6580 t 10 42 45 43 120 116 + 198 t + 5834 t + 5387 t + 14960 t + 720 t - 272 t 115 50 49 46 6 7 8 - 1473 t + 618 t - 10533 t - 8813 t + 35 t + 52 t + 83 t 9 11 13 48 78 118 121 + 139 t + 269 t + 291 t - 20710 t + 19411 t - 72 t + 992 t 119 106 108 12 41 4 5 + 394 t - 4113 t - 3864 t + 359 t - 1560 t + 12 t + 20 t 35 36 14 40 60 15 16 + 8445 t + 5030 t + 61 t - 9132 t - 25740 t - 270 t - 611 t 100 102 31 30 33 27 - 6997 t - 4219 t - 972 t - 2923 t + 6184 t - 1765 t 44 126 128 24 32 29 + 12874 t + 1022 t + 1274 t + 859 t + 2709 t - 4259 t 58 117 38 17 18 57 - 13103 t - 837 t - 6329 t - 857 t - 1016 t - 2537 t 61 21 59 133 105 83 - 33911 t - 430 t - 23742 t + 514 t - 6093 t + 1837 t 132 131 137 142 135 109 + 650 t + 670 t + 170 t + 20 t + 276 t - 3578 t 107 97 98 103 127 55 - 3693 t - 2933 t - 3624 t - 4535 t + 1368 t + 21493 t 47 96 95 93 99 91 - 17215 t - 3616 t - 5533 t - 3891 t - 6916 t - 9920 t 139 140 134 77 75 87 + 64 t + 44 t + 444 t + 16902 t + 7602 t - 1786 t 136 101 130 138 67 85 + 234 t - 3709 t + 760 t + 110 t + 7469 t - 6424 t 143 63 62 65 64 66 + 2 t - 16391 t - 29272 t + 8251 t + 2691 t + 8464 t 122 124 68 69 76 80 + 1012 t + 1300 t + 13165 t + 13310 t + 7045 t + 13983 t 81 79 113 111 112 114 + 7248 t + 19788 t - 2539 t - 2937 t - 2574 t - 1168 t 72 110 84 82 89 92 - 7646 t - 3489 t - 4521 t + 4509 t - 6528 t - 7348 t 90 94 141 73 70 71 - 10653 t - 6316 t + 48 t - 921 t + 3953 t - 5687 t 56 51 54 52 53 88 + 13477 t + 11622 t + 23375 t + 19830 t + 19570 t - 378 t 86 / 39 23 22 74 26 25 - 4763 t ) / (-1 - 20 t - 8 t - 6 t + 36 t - 22 t - 4 t / 34 28 3 20 37 104 42 45 43 + 17 t + 6 t + t + 3 t + 2 t - 2 t + 60 t + 23 t + 24 t 116 115 50 49 46 6 7 8 9 11 - 2 t + t + 68 t + 33 t + 39 t + t + 2 t - t + t + t 13 48 78 106 12 41 5 35 36 14 - t + 14 t + 27 t + t + t + 24 t + 2 t + 9 t + 6 t - t 40 60 16 100 102 31 30 33 - 28 t - 26 t + 3 t + 3 t - t - 26 t - 11 t + 52 t 27 44 24 32 29 58 117 38 17 - 11 t - 2 t + t - 29 t + 7 t - 39 t + t - 15 t - 3 t 18 57 61 21 59 105 83 109 - 4 t - 79 t - 29 t - 14 t - 42 t - 5 t + 13 t - 2 t 107 97 98 103 55 47 96 95 - 7 t + 3 t - 16 t - 13 t - 15 t + 17 t - 4 t + 7 t 93 99 77 75 87 101 67 85 + 3 t - 6 t + 46 t + 56 t + 14 t - 5 t - 11 t + 2 t 63 62 65 64 66 68 69 76 80 - 17 t - 46 t - t + 9 t - 6 t + 5 t + 50 t + 37 t + 13 t 81 79 113 111 112 72 110 84 + 10 t + 58 t + t - 3 t - 2 t + 22 t + t + 15 t 82 89 92 90 94 73 70 71 + 11 t - 6 t - 2 t - 7 t - 10 t + 5 t + 69 t + 31 t 56 51 54 52 53 88 86 - 15 t + 6 t - 21 t - 40 t - 84 t + 8 t - 7 t ) n the asymptotics seems to be roughly, 10.132 1.4106 This took, 566.744, seconds