More Manhattan distinct distances

 

Hello Seq Fans,

We want to form a closed circuit with straight segments at right angle only.

The origin and the start of the circuit is the yellow dot below – usually somewhere on the left:

a) all straight segments must be of distinct length;

b) a straight segment must do a right angle with the previous one (no coming back on the same line);

c) the circuit must be of minimal length – and the lexicographically earliest (explanation follows).

The blue circuit hereunder meets those criteria; the 6-term sequence we can associate to it is 1, 2, 3, 5, 4, 7.

Switching 2 and 5 would form another genuine circuit – but its associated sequence (1, 5, 3, 2, 4, 7) would be lexicographically beaten by the first one: .

The hereunder black circuit is also genuine – but its total length is 24, two units longer than the blue circuit above:

By definition we always come back to the yellow origin when a circuit is completed; from there we can form another circuit – using the smallest available straight segments; we have hereunder the associated sequence 6, 8, 9, 10, 15, 18 for the second circuit:

Note that this second circuit embeds the blue one: this is not an obligation; when one arrives on the yellow dot again after completing the first circuit, one can walk down (south) for the 6-square segment (instead of north here), and complete a symmetrical circuit. The only rules to obey are (a), (b) and (c) at the top of the page.

The third circuit (not presented here) has another associated 6-term sequence: 11, 12, 13, 14, 24, 26.

And the fourth circuit: 16, 17, 19, 20, 35, 37.

The fifth one: 21, 22, 23, 25, 44, 47.

The next one: 27, 28, 29, 30, 56, 58. Etc.

All those circuits, properly concatenated, will produce the sequence S – not in the OEIS, I guess (hope):

S = 1, 2, 3, 5, 4, 7, 6, 8, 9, 10, 15, 18, 11, 12, 13, 14, 24, 26, 16, 17, 19, 20, 35, 37, 21, 22, 23, 25, 44, 47, 27, 28, 29, 30, 56, 58,... 

(The building method of S is easy to understand: always start a new 6-term associated sequence with the smallest integers available not leading to a contradiction).

But is S the lexicographically earliest sequence meeting the criteria (a), (b) and (c)? Remember, we don't want ONE huge circuit to begin with (it's easy to build a first circuit starting with 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... etc.) – we want a concatenation of the shortest possible circuits available at any moment.

Best,

É.

__________

Update, Feb 7th, 2022, two hours after my post

Carole was quick to compute 1002 terms of S:

S = 1, 2, 3, 5, 4, 7, 6, 8, 9, 10, 15, 18, 11, 12, 13, 14, 24, 26, 16, 17, 19, 20, 35, 37, 21, 22, 23, 25, 44, 47, 27, 28, 29, 30, 56, 58, 31, 32, 33, 34, 64, 66, 36, 38, 39, 40, 75, 78, 41, 42, 43, 45, 84, 87, 46, 48, 49, 50, 95, 98, 51, 52, 53, 54, 104, 106, 55, 57, 59, 60, 114, 117, 61, 62, 63, 65, 124, 127, 67, 68, 69, 70, 136, 138, 71, 72, 73, 74, 144, 146, 76, 77, 79, 80, 155, 157, 81, 82, 83, 85, 164, 167, 86, 88, 89, 90, 175, 178, 91, 92, 93, 94, 184, 186, 96, 97, 99, 100, 195, 197, 101, 102, 103, 105, 204, 207, 107, 108, 109, 110, 216, 218, 111, 112, 113, 115, 224, 227, 116, 118, 119, 120, 235, 238, 121, 122, 123, 125, 244, 247, 126, 128, 129, 130, 255, 258, 131, 132, 133, 134, 264, 266, 135, 137, 139, 140, 274, 277, 141, 142, 143, 145, 284, 287, 147, 148, 149, 150, 296, 298, 151, 152, 153, 154, 304, 306, 156, 158, 159, 160, 315, 318, 161, 162, 163, 165, 324, 327, 166, 168, 169, 170, 335, 338, 171, 172, 173, 174, 344, 346, 176, 177, 179, 180, 355, 357, 181, 182, 183, 185, 364, 367, 187, 188, 189, 190, 376, 378, 191, 192, 193, 194, 384, 386, 196, 198, 199, 200, 395, 398, 201, 202, 203, 205, 404, 407, 206, 208, 209, 210, 415, 418, 211, 212, 213, 214, 424, 426, 215, 217, 219, 220, 434, 437, 221, 222, 223, 225, 444, 447, 226, 228, 229, 230, 455, 458, 231, 232, 233, 234, 464, 466, 236, 237, 239, 240, 475, 477, 241, 242, 243, 245, 484, 487, 246, 248, 249, 250, 495, 498, 251, 252, 253, 254, 504, 506, 256, 257, 259, 260, 515, 517, 261, 262, 263, 265, 524, 527, 267, 268, 269, 270, 536, 538, 271, 272, 273, 275, 544, 547, 276, 278, 279, 280, 555, 558, 281, 282, 283, 285, 564, 567, 286, 288, 289, 290, 575, 578, 291, 292, 293, 294, 584, 586, 295, 297, 299, 300, 594, 597, 301, 302, 303, 305, 604, 607, 307, 308, 309, 310, 616, 618, 311, 312, 313, 314, 624, 626, 316, 317, 319, 320, 635, 637, 321, 322, 323, 325, 644, 647, 326, 328, 329, 330, 655, 658, 331, 332, 333, 334, 664, 666, 336, 337, 339, 340, 675, 677, 341, 342, 343, 345, 684, 687, 347, 348, 349, 350, 696, 698, 351, 352, 353, 354, 704, 706, 356, 358, 359, 360, 715, 718, 361, 362, 363, 365, 724, 727, 366, 368, 369, 370, 735, 738, 371, 372, 373, 374, 744, 746, 375, 377, 379, 380, 754, 757, 381, 382, 383, 385, 764, 767, 387, 388, 389, 390, 776, 778, 391, 392, 393, 394, 784, 786, 396, 397, 399, 400, 795, 797, 401, 402, 403, 405, 804, 807, 406, 408, 409, 410, 815, 818, 411, 412, 413, 414, 824, 826, 416, 417, 419, 420, 835, 837, 421, 422, 423, 425, 844, 847, 427, 428, 429, 430, 856, 858, 431, 432, 433, 435, 864, 867, 436, 438, 439, 440, 875, 878, 441, 442, 443, 445, 884, 887, 446, 448, 449, 450, 895, 898, 451, 452, 453, 454, 904, 906, 456, 457, 459, 460, 915, 917, 461, 462, 463, 465, 924, 927, 467, 468, 469, 470, 936, 938, 471, 472, 473, 474, 944, 946, 476, 478, 479, 480, 955, 958, 481, 482, 483, 485, 964, 967, 486, 488, 489, 490, 975, 978, 491, 492, 493, 494, 984, 986, 496, 497, 499, 500, 995, 997, 501, 502, 503, 505, 1004, 1007, 507, 508, 509, 510, 1016, 1018, 511, 512, 513, 514, 1024, 1026, 516, 518, 519, 520, 1035, 1038, 521, 522, 523, 525, 1044, 1047, 526, 528, 529, 530, 1055, 1058, 531, 532, 533, 534, 1064, 1066, 535, 537, 539, 540, 1074, 1077, 541, 542, 543, 545, 1084, 1087, 546, 548, 549, 550, 1095, 1098, 551, 552, 553, 554, 1104, 1106, 556, 557, 559, 560, 1115, 1117, 561, 562, 563, 565, 1124, 1127, 566, 568, 569, 570, 1135, 1138, 571, 572, 573, 574, 1144, 1146, 576, 577, 579, 580, 1155, 1157, 581, 582, 583, 585, 1164, 1167, 587, 588, 589, 590, 1176, 1178, 591, 592, 593, 595, 1184, 1187, 596, 598, 599, 600, 1195, 1198, 601, 602, 603, 605, 1204, 1207, 606, 608, 609, 610, 1215, 1218, 611, 612, 613, 614, 1224, 1226, 615, 617, 619, 620, 1234, 1237, 621, 622, 623, 625, 1244, 1247, 627, 628, 629, 630, 1256, 1258, 631, 632, 633, 634, 1264, 1266, 636, 638, 639, 640, 1275, 1278, 641, 642, 643, 645, 1284, 1287, 646, 648, 649, 650, 1295, 1298, 651, 652, 653, 654, 1304, 1306, 656, 657, 659, 660, 1315, 1317, 661, 662, 663, 665, 1324, 1327, 667, 668, 669, 670, 1336, 1338, 671, 672, 673, 674, 1344, 1346, 676, 678, 679, 680, 1355, 1358, 681, 682, 683, 685, 1364, 1367, 686, 688, 689, 690, 1375, 1378, 691, 692, 693, 694, 1384, 1386, 695, 697, 699, 700, 1394, 1397, 701, 702, 703, 705, 1404, 1407, 707, 708, 709, 710, 1416, 1418, 711, 712, 713, 714, 1424, 1426, 716, 717, 719, 720, 1435, 1437, 721, 722, 723, 725, 1444, 1447, 726, 728, 729, 730, 1455, 1458, 731, 732, 733, 734, 1464, 1466, 736, 737, 739, 740, 1475, 1477, 741, 742, 743, 745, 1484, 1487, 747, 748, 749, 750, 1496, 1498, 751, 752, 753, 755, 1504, 1507, 756, 758, 759, 760, 1515, 1518, 761, 762, 763, 765, 1524, 1527, 766, 768, 769, 770, 1535, 1538, 771, 772, 773, 774, 1544, 1546, 775, 777, 779, 780, 1554, 1557, 781, 782, 783, 785, 1564, 1567, 787, 788, 789, 790, 1576, 1578, 791, 792, 793, 794, 1584, 1586, 796, 798, 799, 800, 1595, 1598, 801, 802, 803, 805, 1604, 1607, 806, 808, 809, 810, 1615, 1618, 811, 812, 813, 814, 1624, 1626, 816, 817, 819, 820, 1635, 1637, 821, 822, 823, 825, 1644, 1647, 827, 828, 829, 830, 1656, 1658, 831, 832, 833, 834, 1664, 1666,...

Merci Carole !

(this seq is now in the OEIS here)

P.-S.

The hereunder circuit was designed by Carole – length 22 like the first circuit of this page, and same integers 1, 2, 4, 5, 3, 7 – but in a different order:

This type of construction (with the equations a + e = c and b + d = f) leads to the hereunder sequence (not submitted to the OEIS as lexicographically not the first, but still interesting):

T = 1, 2, 4, 5, 3, 7, 6, 8, 15, 10, 9, 18, 11, 12, 24, 14, 13, 26, 16, 17, 35, 20, 19, 37, 21, 22, 44, 25, 23, 47, 27, 28, 56, 30, 29, 58, 31, 32, 64, 34, 33, 66, 36, 38, 75, 40, 39, 78, 41, 42, 84, 45, 43, 87, 46, 48, 95, 50, 49, 98, 51, 52, 104, 54, 53, 106, 55, 57, 114, 60, 59, 117, 61, 62, 124, 65, 63, 127, 67, 68, 136, 70, 69, 138, 71, 72, 144, 74, 73, 146, 76, 77, 155, 80, 79, 157, 81, 82, 164, 85, 83, 167, 86, 88, 175, 90, 89, 178, 91, 92, 184, 94, 93, 186, 96, 97, 195, 100, 99, 197, 101, 102, 204, 105, 103, 207, 107, 108, 216, 110, 109, 218, 111, 112, 224, 115, 113, 227, 116, 118, 235, 120, 119, 238, 121, 122, 244, 125, 123, 247, 126, 128, 255, 130, 129, 258, 131, 132, 264, 134, 133, 266, 135, 137, 274, 140, 139, 277, 141, 142, 284, 145, 143, 287, 147, 148, 296, 150, 149, 298, 151, 152, 304, 154, 153, 306, 156, 158, 315, 160, 159, 318, 161, 162, 324, 165, 163, 327, 166, 168, 335, 170, 169, 338, 171, 172, 344, 174, 173, 346, 176, 177, 355, 180, 179, 357, 181, 182, 364, 185, 183, 367, 187, 188, 376, 190, 189, 378, 191, 192, 384, 194, 193, 386, 196, 198, 395, 200, 199, 398, 201, 202, 404, 205, 203, 407, 206, 208, 415, 210, 209, 418, 211, 212, 424, 214, 213, 426, 215, 217, 434, 220, 219, 437, 221, 222, 444, 225, 223, 447, 226, 228, 455, 230, 229, 458, 231, 232, 464, 234, 233, 466, 236, 237, 475, 240, 239, 477, 241, 242, 484, 245, 243, 487, 246, 248, 495, 250, 249, 498, 251, 252, 504, 254, 253, 506, 256, 257, 515, 260, 259, 517, 261, 262, 524, 265, 263, 527, 267, 268, 536, 270, 269, 538, 271, 272, 544, 275, 273, 547, 276, 278, 555, 280, 279, 558, 281, 282, 564, 285, 283, 567, 286, 288, 575, 290, 289, 578, 291, 292, 584, 294, 293, 586, 295, 297, 594, 300, 299, 597, 301, 302, 604, 305, 303, 607, 307, 308, 616, 310, 309, 618, 311, 312, 624, 314, 313, 626, 316, 317, 635, 320, 319, 637, 321, 322, 644, 325, 323, 647, 326, 328, 655, 330, 329, 658, 331, 332, 664, 334, 333, 666, 336, 337, 675, 340, 339, 677, 341, 342, 684, 345, 343, 687, 347, 348, 696, 350, 349, 698, 351, 352, 704, 354, 353, 706, 356, 358, 715, 360, 359, 718, 361, 362, 724, 365, 363, 727, 366, 368, 735, 370, 369, 738, 371, 372, 744, 374, 373, 746, 375, 377, 754, 380, 379, 757, 381, 382, 764, 385, 383, 767, 387, 388, 776, 390, 389, 778, 391, 392, 784, 394, 393, 786, 396, 397, 795, 400, 399, 797, 401, 402, 804, 405, 403, 807, 406, 408, 815, 410, 409, 818, 411, 412, 824, 414, 413, 826, 416, 417, 835, 420, 419, 837, 421, 422, 844, 425, 423, 847, 427, 428, 856, 430, 429, 858, 431, 432, 864, 435, 433, 867, 436, 438, 875, 440, 439, 878, 441, 442, 884, 445, 443, 887, 446, 448, 895, 450, 449, 898, 451, 452, 904, 454, 453, 906, 456, 457, 915, 460, 459, 917, 461, 462, 924, 465, 463, 927, 467, 468, 936, 470, 469, 938, 471, 472, 944, 474, 473, 946, 476, 478, 955, 480, 479, 958, 481, 482, 964, 485, 483, 967, 486, 488, 975, 490, 489, 978, 491, 492, 984, 494, 493, 986, 496, 497, 995, 500, 499, 997, 501, 502, 1004, 505, 503, 1007, 507, 508, 1016, 510, 509, 1018, 511, 512, 1024, 514, 513, 1026, 516, 518, 1035, 520, 519, 1038, 521, 522, 1044, 525, 523, 1047, 526, 528, 1055, 530, 529, 1058, 531, 532, 1064, 534, 533, 1066, 535, 537, 1074, 540, 539, 1077, 541, 542, 1084, 545, 543, 1087, 546, 548, 1095, 550, 549, 1098, 551, 552, 1104, 554, 553, 1106, 556, 557, 1115, 560, 559, 1117, 561, 562, 1124, 565, 563, 1127, 566, 568, 1135, 570, 569, 1138, 571, 572, 1144, 574, 573, 1146, 576, 577, 1155, 580, 579, 1157, 581, 582, 1164, 585, 583, 1167, 587, 588, 1176, 590, 589, 1178, 591, 592, 1184, 595, 593, 1187, 596, 598, 1195, 600, 599, 1198, 601, 602, 1204, 605, 603, 1207, 606, 608, 1215, 610, 609, 1218, 611, 612, 1224, 614, 613, 1226, 615, 617, 1234, 620, 619, 1237, 621, 622, 1244, 625, 623, 1247, 627, 628, 1256, 630, 629, 1258, 631, 632, 1264, 634, 633, 1266, 636, 638, 1275, 640, 639, 1278, 641, 642, 1284, 645, 643, 1287, 646, 648, 1295, 650, 649, 1298, 651, 652, 1304, 654, 653, 1306, 656, 657, 1315, 660, 659, 1317, 661, 662, 1324, 665, 663, 1327, 667, 668, 1336, 670, 669, 1338, 671, 672, 1344, 674, 673, 1346, 676, 678, 1355, 680, 679, 1358, 681, 682, 1364, 685, 683, 1367, 686, 688, 1375, 690, 689, 1378, 691, 692, 1384, 694, 693, 1386, 695, 697, 1394, 700, 699, 1397, 701, 702, 1404, 705, 703, 1407, 707, 708, 1416, 710, 709, 1418, 711, 712, 1424, 714, 713, 1426, 716, 717, 1435, 720, 719, 1437, 721, 722, 1444, 725, 723, 1447, 726, 728, 1455, 730, 729, 1458, 731, 732, 1464, 734, 733, 1466, 736, 737, 1475, 740, 739, 1477, 741, 742, 1484, 745, 743, 1487, 747, 748, 1496, 750, 749, 1498, 751, 752, 1504, 755, 753, 1507, 756, 758, 1515, 760, 759, 1518, 761, 762, 1524, 765, 763, 1527, 766, 768, 1535, 770, 769, 1538, 771, 772, 1544, 774, 773, 1546, 775, 777, 1554, 780, 779, 1557, 781, 782, 1564, 785, 783, 1567, 787, 788, 1576, 790, 789, 1578, 791, 792, 1584, 794, 793, 1586, 796, 798, 1595, 800, 799, 1598, 801, 802, 1604, 805, 803, 1607, 806, 808, 1615, 810, 809, 1618, 811, 812, 1624, 814, 813, 1626, 816, 817, 1635, 820, 819, 1637, 821, 822, 1644, 825, 823, 1647, 827, 828, 1656, 830, 829, 1658, 831, 832, 1664, 834, 833, 1666, ...