************************************* The first , 30, terms of the counting sequence for binary words avoiding the set of patterns, {[1, 0, 1, 0]}, with spacings <= , 0, 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] 2 t + 1 The generating function is, ------------------------- 3 4 2 -2 t + t + 1 - 2 t + t n the asymptotics seems to be roughly, 1.1892 1.8832 The first , 30, terms of the counting sequence for binary words avoiding the set of patterns, {[1, 0, 1, 0]}, with spacings <= , 1, 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] 2 4 5 7 8 6 10 The generating function is, (-t - 1 - 3 t - 2 t + 2 t + 3 t - 3 t + 2 t 11 12 9 14 / 2 3 4 5 6 - 2 t + t + 2 t + t ) / (2 t - 1 - t + 2 t - 4 t + 4 t + t / 9 10 8 12 14 13 - 6 t + 3 t - 3 t + 4 t + t - 2 t ) n the asymptotics seems to be roughly, 2.1199 1.7168 The first , 30, terms of the counting sequence for binary words avoiding the set of patterns, {[1, 0, 1, 0]}, with spacings <= , 2, 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] 16 22 2 4 17 15 The generating function is, (1 + t + 2 t + t + 3 t - 8 t - 2 t 20 19 32 34 5 7 8 9 6 10 - 8 t - 6 t - 8 t + 4 t + 2 t + 4 t + 7 t + 4 t + 9 t - 4 t 11 13 14 12 23 21 30 25 26 - 4 t - 8 t - t - 6 t + 4 t + 2 t + 4 t + 8 t + 3 t 18 / 16 24 22 2 3 4 - 21 t ) / (1 - 2 t + 3 t + 19 t - 9 t + t - 2 t + 4 t / 17 15 20 19 32 5 7 8 9 6 - 10 t - 10 t + 4 t - 4 t + 3 t - 4 t - 6 t + t - 8 t + 5 t 10 11 13 14 12 23 21 30 29 + 13 t - 2 t - 6 t + 8 t + 9 t - 2 t - 16 t - 2 t - 10 t 25 28 26 18 + 6 t - t + 7 t + 7 t ) n the asymptotics seems to be roughly, 3.4999 1.6225 The first , 30, terms of the counting sequence for binary words avoiding the set of patterns, {[1, 0, 1, 0]}, with spacings <= , 3, 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] 65 76 16 The generating function is, (1 - 67550 t + 2 t - 6728 t + 495 t 24 22 56 58 2 3 4 - 3008 t - 745 t - 62176 t - 74338 t + 4 t + 6 t + 11 t 71 73 74 67 69 17 - 37440 t - 21122 t - 14188 t - 60356 t - 51152 t + 414 t 15 20 19 79 81 32 34 + 548 t - 49 t + 178 t - 6242 t - 4972 t - 28571 t - 30286 t 82 52 75 5 7 8 9 6 - 3601 t + 968 t - 9304 t + 18 t + 54 t + 89 t + 138 t + 34 t 10 72 54 11 13 14 12 + 201 t - 29199 t - 36786 t + 286 t + 478 t + 542 t + 399 t 42 23 66 60 63 77 + 59224 t - 1508 t - 63982 t - 78102 t - 74316 t - 5966 t 80 21 87 83 30 29 - 5976 t - 316 t + 298 t - 2116 t - 22107 t - 18240 t 25 78 86 85 28 26 18 - 5154 t - 6089 t + 249 t - 66 t - 14664 t - 8079 t + 316 t 84 53 33 31 35 89 - 864 t - 19138 t - 30230 t - 25738 t - 28462 t + 106 t 91 92 68 70 27 93 88 + 22 t + 6 t - 56202 t - 44910 t - 11220 t + 4 t + 197 t 90 36 38 37 40 39 + 51 t - 23603 t - 3779 t - 15584 t + 26042 t + 10190 t 41 64 55 57 59 61 + 42456 t - 71111 t - 51266 t - 69858 t - 76910 t - 78054 t 62 44 46 43 45 47 - 76681 t + 86551 t + 94992 t + 74566 t + 93760 t + 90526 t 49 48 50 51 / 65 + 64824 t + 80217 t + 44780 t + 23014 t ) / (1 - 1036 t / 76 16 24 22 56 58 3 71 - 464 t + 17 t + 69 t + 61 t - 87 t + 257 t - 2 t - 1124 t 73 74 67 69 17 15 20 - 630 t - 528 t - 1136 t - 1322 t + 46 t + 18 t + 38 t 19 79 81 32 34 82 52 + 82 t - 270 t - 134 t - 540 t - 392 t - 83 t - 1138 t 75 5 7 8 9 6 10 72 54 - 468 t - 2 t - 2 t + 4 t - 4 t + t - t - 864 t - 1125 t 11 13 14 12 42 23 66 - 10 t - 22 t - 10 t - 20 t + 1628 t + 98 t - 995 t 60 63 77 80 21 87 83 - 493 t - 1200 t - 442 t - 186 t + 100 t - 36 t - 74 t 30 29 25 78 86 85 28 - 617 t - 532 t + 84 t - 376 t - 65 t - 74 t - 345 t 26 18 84 53 33 31 35 - 26 t + 38 t - 87 t - 1218 t - 576 t - 732 t - 382 t 89 91 68 70 27 88 90 36 + 10 t + 4 t - 1313 t - 1312 t - 188 t - 7 t + 6 t - 189 t 38 37 40 39 41 64 55 + 261 t - 122 t + 1146 t + 694 t + 1474 t - 1084 t - 702 t 57 59 61 62 44 46 43 + 172 t - 64 t - 1012 t - 1161 t + 1105 t + 536 t + 1586 t 45 47 49 48 50 51 + 896 t + 440 t + 136 t + 220 t - 316 t - 742 t ) n the asymptotics seems to be roughly, 7.8374 1.5183 The first , 30, terms of the counting sequence for binary words avoiding the set of patterns, {[1, 0, 1, 0]}, with spacings <= , 4, are [2, 4, 8, 15, 28, 53, 92, 157, 270, 435, 704, 1142, 1734, 2646, 4018, 5798, 8456, 12313, 17654, 25673, 37670, 54955, 81068, 120294, 177054, 262245, 387632, 568241, 833094, 1215764] 65 76 16 24 The generating function is, - (-1 - 111088 t + 58569 t + 116 t + 145 t 22 56 58 2 4 71 73 + 16 t + 82714 t + 60262 t - t - 3 t - 122444 t - 42956 t 74 146 164 167 134 133 - 4684 t + 103 t + 16 t + 12 t - 1062 t - 790 t 136 138 156 158 171 67 + 587 t + 1119 t - 77 t + 4 t + 2 t - 128864 t 69 17 15 20 19 157 137 - 139272 t + 82 t + 4 t + 63 t + 88 t - 194 t + 726 t 135 139 140 142 141 144 - 94 t + 870 t - 200 t + 765 t + 318 t + 136 t 143 145 152 155 154 168 159 + 638 t - 356 t + 86 t + 210 t + 212 t - 12 t + 32 t 161 163 165 166 148 150 147 - 22 t + 68 t - 8 t + 4 t - 242 t + 672 t + 154 t 149 151 153 79 81 32 + 42 t + 548 t + 82 t + 188688 t + 206610 t + 1735 t 34 82 52 75 5 7 8 - 1011 t + 188495 t + 38899 t + 43842 t - 2 t - 4 t - 15 t 9 6 10 72 54 11 13 14 - 14 t - 9 t - 20 t - 93151 t + 61364 t - 28 t + 4 t - 39 t 12 42 23 66 60 63 - 56 t - 17917 t - 178 t - 138378 t + 23893 t - 35606 t 77 169 160 162 80 21 + 93918 t - 16 t - 56 t + 68 t + 191302 t - 96 t 87 83 30 29 25 78 - 26028 t + 154360 t + 3685 t + 3000 t + 194 t + 154282 t 86 98 85 28 26 18 + 5655 t - 12725 t + 50862 t + 2027 t + 773 t + 105 t 84 100 53 111 123 112 + 109657 t + 25863 t + 55856 t - 3730 t - 7850 t - 5525 t 33 31 35 89 91 92 + 1112 t + 3026 t - 1810 t - 92846 t - 130646 t - 123834 t 68 70 27 122 121 93 - 144380 t - 146205 t + 1552 t - 11976 t - 9648 t - 102028 t 88 90 36 38 37 40 - 79071 t - 98515 t - 2085 t - 4439 t - 3292 t - 11097 t 39 41 120 102 119 127 - 7836 t - 14068 t - 3990 t + 48853 t - 3488 t - 970 t 114 116 117 118 113 115 - 13824 t + 2638 t + 3404 t - 1991 t - 4250 t - 11912 t 124 64 132 131 129 128 - 960 t - 76221 t + 171 t - 1220 t - 428 t - 57 t 55 57 59 61 62 126 + 83712 t + 77248 t + 47940 t + 13370 t - 16283 t - 1773 t 44 46 43 45 47 49 - 17862 t - 19545 t - 20602 t - 19656 t - 13034 t + 6800 t 48 50 51 130 110 109 - 7383 t + 15459 t + 28040 t - 1284 t + 11284 t + 25158 t 96 94 99 95 97 107 - 96550 t - 115025 t + 9672 t - 117338 t - 61330 t + 36004 t 108 125 106 105 101 + 35585 t + 852 t + 38852 t + 55232 t + 52714 t 104 103 / 65 76 16 + 55726 t + 37376 t ) / (1 + 1522 t - 2 t - 45 t + 42 t / 24 22 56 58 2 3 4 71 73 + 83 t + 6 t + t - 358 t + t - 2 t + 4 t - 540 t + 1754 t 74 67 69 17 15 20 19 - 2586 t + 2850 t - 114 t - 6 t - 32 t + 25 t - 40 t 79 81 32 34 82 52 75 + 1868 t + 1314 t + 110 t + 62 t - 203 t - 110 t + 254 t 5 7 8 9 6 10 72 54 11 - 4 t - 6 t + 9 t - 14 t + 5 t + 17 t - 1433 t - 298 t - 18 t 13 14 12 42 23 66 60 63 - 18 t + 14 t + 15 t + 383 t - 48 t + 7 t - 606 t + 78 t 77 80 21 87 83 30 29 - 2458 t - 3431 t - 42 t + 2324 t - 1776 t + 95 t - 86 t 25 78 86 98 85 28 26 - 76 t - 142 t - 2178 t - 763 t + 1110 t + 17 t + 40 t 18 84 100 53 111 123 112 + 49 t + 1403 t - 1237 t - 560 t + 250 t + 20 t - 264 t 33 31 35 89 91 92 68 - 172 t - 100 t - 198 t - 864 t + 164 t - 1422 t - 42 t 70 27 122 121 93 88 90 + 966 t - 60 t - 36 t + 18 t + 2426 t - 1476 t + 2623 t 36 38 37 40 39 41 120 + 204 t + 388 t - 138 t + 677 t - 74 t - 248 t + 68 t 102 119 127 114 116 117 118 + 934 t - 8 t + 8 t + 148 t - 16 t + 164 t - 40 t 113 115 124 64 55 57 59 + 46 t - 78 t + 16 t + 1150 t - 1116 t - 1146 t + 608 t 61 62 126 44 46 43 45 + 366 t + 569 t - 8 t + 470 t + 578 t - 630 t - 676 t 47 49 48 50 51 110 109 - 244 t - 816 t + 786 t + 796 t - 628 t - 264 t - 364 t 96 94 99 95 97 107 108 + 1969 t - 1355 t + 1230 t - 96 t - 420 t - 48 t + 470 t 106 105 101 104 103 - 985 t + 630 t + 264 t - 392 t - 824 t ) n the asymptotics seems to be roughly, 13.194 1.4614 This took, 1505.004, seconds