Palindromes chain
Here is the idea:
a] start the sequence S with a(1)=0
b] a(n)
is the smallest integer not yet present in S that can be linked to S;
c] a
link is a palindromic
pattern involving one or more digits of the new term and one or more digits of
the last term(s) of S.
Explanation:
S=0,10,1,11,12,2,13,3,14,4,15,5,...
a(2)=10 because there is the palindromic link 0,10 between a(1)
and a(2) 10 being the smallest integer allowing such a link;
a(3)=1 because there is the palindromic link 10,1 between a(2)
and a(3) 1 being the smallest integer allowing such a link;
a(4)=11 because there is the palindromic link 1,1 between a(3)
and a(4) 11 being the smallest integer allowing such a link;
a(5)=12 because there is the palindromic link 1,1 between a(4)
and a(5) 12 being the smallest integer allowing such a link;
a(6)=2 because there is the palindromic link 2,2 between a(5)
and a(6) 2 being the smallest integer allowing such a link;
a(7)=13 because there is the palindromic link 12,2,1 involving
a(5), a(6) and a(7) 13 being the smallest integer allowing such a link;
a(8)=3 because there is the palindromic link 3,3 between a(7)
and a(8) 3 being the smallest integer allowing such a link;
a(9)=14 because there is the palindromic link 13,3,1 involving
a(7), a(8) and a(9) 14 being the smallest integer allowing such a link;
... etc.
__________
I guess S goes like this (but Im not 100% sure this has been
computed by hand):
S = 0, 10, 1, 11, 12, 2, 13, 3, 14, 4, 15, 5, 16, 6, 17, 7, 18, 8, 19, 9,
29, 20, 21, 22, 23, 24, 25, 26, 27, 28, 38, 30, 31, 32, 33, 34, 35, 36, 37, 39,
49, 40, 41, 42, 43, 44, 45, 46, 47, 48, 58, 50, 51, 52, 53, 54, 55, 56, 57, 59,
69, 60, 61, 62, 63, 64, 65, 66, 67, 68, 78, 70, 71, 72, 73, 74, 75, 76, 77, 79,
89, 80, 81, 82, 83, 84, 85, 86, 87, 88, 98, 90, 91, 92, 93, 94, 95, 96, 97, 99,
190, 100, 101, 102, 112, 103, 113, 104, 114, 105, 115, 106, 116, 107, 117, 108,
118, 109,
I guess S is a permutation of the nonnegative integers.
Best,
Ι.
__________
Jean-Marc Falcoz was
quick to compute S and propose a few beautiful graphs
(217 terms, 1000 terms and 2341 terms):
(The first 2341
terms of S; smallest missing integer: 1994)
{0, 10, 1, 11, 12, 2, 13, 3, 14, 4, 15, 5, 16,
6, 17, 7, 18, 8, 19, 9, 29, 20, 21, 22, 23, 24, 25, 26, 27, 28, 38, 30, 31, 32,
33, 34, 35, 36, 37, 39, 49, 40, 41, 42, 43, 44, 45, 46, 47, 48, 58, 50, 51, 52,
53, 54, 55, 56, 57, 59, 69, 60, 61, 62, 63, 64, 65, 66, 67, 68, 78, 70, 71, 72,
73, 74, 75, 76, 77, 79, 89, 80, 81, 82, 83, 84, 85, 86, 87, 88, 98, 90, 91, 92,
93, 94, 95, 96, 97, 99, 119, 100, 101, 102, 112, 103, 113, 104, 114, 105, 115, 106,
116, 107, 117, 108, 118, 109, 190, 110, 111, 120, 200, 201, 121, 122, 123, 130,
202, 124, 140, 203, 131, 125, 150, 204, 141, 126, 160, 205, 151, 127, 170, 206,
161, 128, 180, 207, 171, 129, 191, 132, 208, 181, 133, 134, 142, 209, 192, 210,
135, 152, 211, 136, 162, 212, 137, 172, 213, 138, 182, 214, 139, 193, 143, 223,
215, 144, 145, 153, 230, 220, 216, 146, 163, 231, 147, 173, 232, 217, 148, 183,
233, 218, 149, 194, 154, 224, 219, 155, 156, 164, 240, 300, 301, 157, 174, 241,
158, 184, 242, 221, 159, 195, 165, 225, 222, 226, 166, 167, 175, 250, 302, 227,
176, 168, 185, 251, 169, 196, 177, 178, 186, 260, 303, 234, 243, 228, 187, 179,
197, 188, 189, 198, 199, 229, 235, 252, 236, 261, 310, 304, 244, 237, 270, 305,
253, 238, 280, 306, 262, 239, 290, 307, 271, 311, 308, 281, 312, 245, 254, 246,
263, 309, 291, 313, 314, 247, 272, 248, 282, 249, 292, 255, 256, 264, 334, 315,
257, 273, 316, 265, 258, 283, 317, 274, 340, 330, 318, 284, 341, 319, 293, 320,
259, 294, 342, 266, 267, 275, 335, 321, 268, 285, 350, 400, 401, 331, 322, 269,
295, 351, 410, 402, 276, 336, 323, 277, 278, 286, 360, 403, 324, 279, 296, 361,
411, 404, 343, 325, 287, 337, 326, 288, 289, 297, 370, 405, 352, 298, 338, 327,
299, 339, 328, 380, 406, 362, 329, 390, 407, 371, 412, 332, 333, 344, 345, 353,
346, 363, 347, 372, 420, 408, 381, 413, 348, 382, 421, 414, 349, 391, 415, 354,
409, 392, 422, 416, 364, 417, 373, 355, 356, 365, 357, 374, 418, 383, 358, 384,
419, 393, 359, 394, 423, 366, 367, 375, 445, 424, 425, 450, 440, 426, 368, 385,
451, 441, 427, 376, 369, 395, 452, 428, 386, 446, 429, 396, 460, 500, 501, 510,
502, 442, 430, 377, 378, 387, 379, 397, 388, 389, 398, 399, 449, 431, 511, 503,
432, 520, 504, 433, 434, 435, 453, 436, 461, 512, 521, 505, 454, 437, 447, 438,
448, 439, 490, 506, 462, 522, 507, 470, 508, 480, 509, 491, 513, 443, 444, 455,
456, 463, 530, 550, 514, 457, 471, 515, 458, 481, 516, 464, 459, 492, 523, 531,
517, 472, 524, 465, 518, 482, 525, 519, 493, 532, 526, 466, 467, 473, 533, 527,
474, 468, 483, 534, 469, 494, 475, 528, 484, 476, 556, 529, 495, 535, 536, 560,
600, 601, 551, 537, 477, 478, 485, 538, 486, 561, 602, 552, 539, 496, 562, 603,
553, 540, 479, 497, 487, 557, 541, 488, 489, 498, 499, 559, 542, 620, 604, 543,
630, 605, 544, 545, 546, 563, 606, 564, 547, 570, 607, 548, 558, 549, 590, 608,
580, 609, 554, 555, 565, 566, 567, 571, 610, 660, 611, 612, 621, 613, 631, 614,
640, 700, 701, 615, 568, 581, 616, 569, 591, 617, 572, 622, 618, 582, 623, 632,
619, 592, 624, 641, 661, 625, 573, 633, 626, 627, 574, 642, 628, 583, 634, 643,
629, 593, 635, 575, 576, 636, 637, 577, 578, 584, 644, 638, 585, 579, 594, 645,
586, 639, 595, 587, 667, 646, 647, 670, 702, 662, 648, 588, 589, 596, 649, 597,
671, 703, 663, 650, 598, 668, 651, 599, 669, 652, 720, 704, 664, 653, 730, 705,
654, 740, 706, 655, 656, 657, 672, 707, 673, 708, 680, 709, 690, 710, 770, 711,
712, 721, 713, 731, 714, 741, 715, 658, 681, 716, 659, 691, 717, 674, 718, 682,
722, 719, 692, 723, 732, 724, 742, 725, 665, 666, 675, 726, 676, 677, 678, 683,
733, 727, 679, 693, 734, 743, 728, 684, 744, 729, 694, 745, 750, 800, 801, 771,
735, 751, 810, 802, 772, 736, 685, 752, 820, 803, 737, 738, 686, 687, 739, 695,
753, 773, 746, 688, 689, 696, 697, 747, 748, 778, 749, 698, 780, 804, 774, 754,
840, 805, 755, 756, 699, 779, 757, 758, 781, 806, 760, 807, 759, 790, 808, 782,
809, 791, 811, 812, 821, 813, 830, 880, 814, 841, 815, 775, 761, 816, 762, 822,
817, 763, 831, 818, 783, 819, 792, 823, 832, 824, 842, 825, 850, 900, 901, 881,
826, 764, 843, 833, 827, 765, 851, 910, 902, 828, 784, 829, 793, 834, 844, 835,
852, 882, 836, 766, 767, 768, 785, 837, 769, 794, 845, 853, 838, 786, 776, 777,
787, 788, 789, 795, 854, 846, 860, 903, 839, 796, 861, 911, 904, 847, 797, 798,
848, 849, 799, 889, 855, 856, 862, 920, 905, 857, 870, 906, 863, 883, 858, 859,
890, 907, 871, 912, 921, 908, 864, 884, 865, 885, 866, 867, 872, 922, 909, 891,
913, 930, 990, 914, 940, 1000, 1001, 915, 950, 1002, 923, 931, 916, 868, 869, 892,
917, 873, 932, 924, 941, 918, 874, 942, 925, 951, 919, 893, 926, 886, 875, 952,
927, 876, 960, 1003, 933, 928, 877, 878, 879, 894, 929, 895, 934, 943, 935, 953,
936, 961, 991, 937, 887, 888, 896, 938, 897, 939, 898, 899, 944, 945, 954, 946,
962, 992, 947, 970, 1004, 948, 980, 1005, 955, 949, 956, 963, 993, 957, 971, 1006,
964, 994, 958, 981, 1007, 972, 1012, 1008, 982, 1112, 1009, 959, 965, 995, 966,
967, 973, 1013, 1010, 1011, 1014, 1015, 1016, 968, 983, 1113, 1017, 974, 1114, 1018,
984, 1140, 1019, 969, 975, 1115, 1020, 1021, 1022, 1120, 1023, 1130, 1024, 1141,
1025, 1150, 1026, 996, 976, 1116, 1027, 977, 978, 985, 1151, 1028, 986, 1160, 1029,
979, 987, 997, 988, 989, 998, 999, 1119, 1030, 1031, 1032, 1121, 1033, 1131, 1034,
1142, 1122, 1035, 1152, 1123, 1132, 1036, 1161, 1037, 1117, 1038, 1118, 1039, 1190,
1040, 1041, 1042, 1124, 1143, 1133, 1043, 1134, 1144, 1044, 1145, 1153, 1135, 1045,
1154, 1146, 1162, 1125, 1155, 1046, 1163, 1136, 1047, 1170, 1048, 1180, 1049, 1191,
1050, 1051, 1052, 1126, 1164, 1147, 1171, 1053, 1137, 1172, 1127, 1054, 1148, 1181,
1055, 1156, 1165, 1056, 1166, 1057, 1173, 1138, 1182, 1128, 1058, 1183, 1139, 1192,
1129, 1059, 1193, 1213, 1060, 1061, 1062, 1200, 1063, 1300, 1064, 1149, 1194, 1065,
1157, 1174, 1214, 1066, 1167, 1175, 1158, 1184, 1314, 1067, 1176, 1168, 1185, 1159,
1195, 1068, 1186, 1169, 1196, 1069, 1197, 1177, 1070, 1071, 1072, 1201, 1073, 1301,
1074, 1400, 1075, 1215, 1076, 1216, 1077, 1178, 1187, 1078, 1188, 1079, 1198, 1189,
1080, 1081, 1082, 1202, 1083, 1302, 1203, 1084, 1401, 1085, 1315, 1086, 1316, 1087,
1179, 1199, 1088, 1218, 1089, 1219, 1090, 1091, 1092, 1204, 1402, 1093, 1303, 1094,
1403, 1304, 1095, 1415, 1096, 1416, 1097, 1217, 1098, 1318, 1099, 1319, 1100, 1101,
1102, 1205, 1500, 1103, 1305, 1501, 1104, 1404, 1105, 1502, 1206, 1516, 1106, 1600,
1107, 1317, 1108, 1418, 1109, 1419, 1110, 1111, 1207, 1417, 1208, 1518, 1209, 1519,
1210, 1211, 1212, 1220, 1310, 1221, 1222, 1223, 1306, 1601, 1224, 1405, 1503, 1307,
1517, 1225, 1504, 1406, 1602, 1226, 1603, 1308, 1618, 1227, 1617, 1228, 1718, 1229,
1619, 1230, 1410, 1231, 1232, 1233, 1309, 1719, 1234, 1407, 1700, 1510, 1235, 1505,
1236, 1604, 1408, 1800, 1610, 1237, 1701, 1238, 1801, 1239, 1819, 1240, 1710, 1241,
1242, 1243, 1311, 1244, 1409, 1900, 1810, 1245, 1506, 1605, 1246, 1606, 1247, 1702,
1248, 1802, 1249, 1901, 1250, 1910, 1251, 1252, 1253, 1312, 1254, 1411, 1255, 1507,
1703, 1313, 1256, 1607, 1704, 1412, 1257, 1705, 1508, 1803, 1320, 2000, 2001, 1258,
1804, 1413, 1259, 1902, 1260, 2002, 1261, 1262, 1263, 1321, 1264, 1414, 1265, 1509,
1903, 1322, 1266, 1608, 1805, 1511, 1267, 1706, 1609, 1904, 1420, 2003, 1323, 1268,
1806, 1611, 1269, 1905, 1512, 1270, 2004, 1421, 1271, 1272, 1273, 1324, 1422, 1274,
1423, 1325, 1513, 1275, 1514, 1276, 1612, 1277, 1707, 1278, 1807, 1708, 1279, 1906,
1613, 1280, 2005, 1515, 1281, 1282, 1283, 1326, 1614, 1284, 1424, 1285, 1520, 2006,
1615, 1286, 1616, 1287, 1709, 1907, 1288, 1808, 1289, 1908, 1809, 1290, 2007, 1711,
1291, 1292, 1293, 1327, 1712, 1294, 1425, 1521, 1295, 1522, 1296, 1620, 2008, 1811,
1297, 1713, 1298, 1812, 1299, 1909, 1328, 1813, 1329, 1911, 1330, 2009, 1912, 1331,
1332, 2010, 1333, 1334, 1426, 1621, 1335, 1523, 1336, 1622, 2011, 1337, 1714, 1338,
1814, 1339, 1913, 1340, 2012, 1341, 1342, 2013, 1343, 1344, 1427, 1715, 1345, 1524,
1428, 1815, 1346, 1623, 1347, 1716, 1348, 1816, 1349, 1914, 1350, 2014, 1351, 1352,
2015, 1353, 1354, 1429, 1915, 1355, 1525, 1356, 1624, 1430, 2016, 1357, 1717, 1358,
1817, 1359, 1916, 1360, 2017, 1361, 1362, 2018, 1363, 1364, 1431, 1365, 1526, 1625,
1366, 1626, 1367, 1720, 2019, 1368, 1818, 1369, 1917, 1370, 2020, 2021, 1371, 1372,
2022, 2023, 1373, 1374, 1432, 2024, 1433, 1375, 1527, 1721, 1376, 1627, 1722, 2025,
1528, 1820, 2026, 1628, 1821, 1377, 1723, 1378, 1822, 2027, 1724, 1434, 1379, 1918,
1380, 2028, 1823, 1381, 1382, 2029, 1919, 1383, 1384, 1435, 1529, 1920, 2030, 2031,
1385, 1530, 2032, 2033, 1386, 1629, 1921, 1387, 1725, 1531, 1388, 1824, 1436, 1630,
2034, 1437, 1726, 1631, 1389, 1922, 2035, 1532, 2036, 1632, 2037, 1727, 1390, 2038,
1825, 1533, 1391, 1392, 2039, 1923, 1393, 1394, 1438, 1826, 1633, 1395, 1534, 1439,
1924, 1440, 2040, 2041, 1396, 1634, 1441, 1397, 1728, 1827, 1398, 1828, 1399, 1925,
1535, 1442, 2042, 2043, 2123, 2044, 1443, 2223, 2045, 1536, 1635, 1444, 1445, 1537,
1729, 1926, 1636, 1446, 1637, 1730, 2046, 1638, 1829, 1927, 1731, 1447, 1732, 2047,
1733, 2230, 2048, 1830, 2049, 1928, 1831, 1448, 1832, 2050, 2051, 1449, 1929, 1450,
2052, 2053, 2231, 1451, 1452, 2054, 1453, 2232, 2055, 1538, 1833, 2233, 2056, 1639,
1930, 2057, 1734, 1454, 1455, 1539, 1931, 1456, 1640, 2058, 1834, 1457, 1735, 1540,
2059, 1932, 2060, 2061, 1458, 1835, 1541, 1459, 1933, 2234, 1460, 2062, 2063, 2235,
1542, 2064, 1461, 1462, 2065, 1543, 2236, 1641, 1463, 2237, 1736, 1642, 2066, 1643,
2238, 1836, 1644, 1464, 1465, 1544, 1466, 1645, 1545, 1467, 1737, 1468, 1837, 1738,
1469, 1934, 1470, 2067, 1739, 1935, 1546, 1646, 1471, 1472, 2068, 1838, 1473, 2239,
1936, 1647, 1740, 2069, 1937, 1741, 1474, 1475, 1547, 1742, 2070, 2071, 1476, 1648,
1839, 1938, 1477, 1743, 2300, 2072, 2073, 2301, 1478, 1840, 2074, 1479, 1939, 1480,
2075, 1548, 1841, 1481, 1482, 2076, 1649, 1940, 2077, 1744, 1483, 2302, 2078, 1842,
2079, 1941, 1484, 1485, 1549, 1942, 2080, 2081, 1486, 1650, 2082, 2083, 2303, 2084,
1487, 1745, 1550, 2085, 1551, 1488, 1843, 2304, 1489, 1943, 2305, 1552, 2086, 1651,
1490, 2087, 1746, 1652, 2088, 1844, 1491, 1492, 2089, 1944, 1493, 2306, 1653, 2307,
1747, 1494, 1495, 1553, 2308, 1845, 1554, 1496, 1654, 1497, 1748, 1846, 1655, 1555,
1498, 1847, 1749, 1945, 1556, 1656, 1499, 1946, 1657, 1750, 2090, 2091, 1557, 1751,
1558, 1848, 1559, 1947, 1752, 2092, 2093, 2309, 1948, 1849, 1560, 2094, 2124, 2095,
1561, 1562, 2096, 1658, 1850, 2097, 1753, 2310, 1563, 2311, 1564, 2224, 2098, 1851,
1565, 1566, 1659, 1949, 1567, 1754, 2240, 2099, 1950, 2120, 2100, 1568, 1852, 2101,
1569, 1951, 1570, 2200, 2102, 2103, 2312, 1571, 1572, 2104, 2241, 1573, 2313, 1574,
2242, 2105, 1575, 1576, 1660, 2201, 1577, 1755, 1578, 1853, 2314, 1579, 1952, 2106,
1661, 1580, 2202, 2107, 1756, 1662, 2108, 1854, 2243, 2315, 1581, 1582, 2109, 1953,
2316, 1583, 2317, 1584, 2244, 2110, 1585, 1586, 1663, 2318, 1587, 1757, 1588, 1855,
1589, 1954, 2245, 1590, 2203, 2319, 1591, 1592, 2111, 1593, 2320, 2112, 1594, 2246,
1664, 2247, 1758, 1856, 1665, 1595, 1596, 1666, 1597, 1759, 1955, 1598, 1857, 1760,
2204, 2248, 1858, 1599, 1956, 1667, 1761, 1668, 1859, 1957, 1762, 2113, 1669, 1958,
1860, 2205, 2125, 2114, 1670, 2206, 1671, 1672, 2115, 1673, 2321, 1674, 2249, 1959,
1675, 2225, 2116, 1676, 1677, 1763, 2322, 2117, 1678, 1861, 1679, 1960, 2207, 1764,
2324, 2118, 1680, 2208, 1862, 2119, 1681, 1682, 2121, 1683, 2323, 2122, 1684, 2400,
2209, 1961, 1685, 2250, 2220, 2126, 1686, 1687, 1765, 2251, 1688, 1863, 2325, 2127,
1766, 1689, 1962, 2128, 1864, 2401, 1690, 2420, 2129, 1963, 2326, 1691, 1692, 2130,
2520, 2131, 1693, 2327, 1767, 1694, 2402, 2132, 2133, 1695, 2252, 2134, 2403, 2328,
1865, 2253, 2329, 1964, 2404, 2135, 2254, 2405, 2255, 2136, 1696, 1697, 1768, 1866,
1698, 1867, 1769, 1965, 2256, 1699, 1966, 2226, 2137, 1770, 2620, 2138, 1868, 1771,
1772, 2139, 1967, 1773, 2330, 2720, 2140, 2820, 2141, 1774, 2406, 2260, 2920, 2142,
2143, 2331, 1775, 2257, 1776, 2261, 1777, 1778, 1869, 1968, 1779, 1969, 1780, 3000,
3001, 1781, 1782, 2144, 1783, 2332, 2145, 2258, 1870, 3002, 2146, 2262, 2147, 1784,
2407, 1785, 2259, 1970, 3003, 2333, 2148, 1871, 1786, 2263, 2334, 2408, 1872, 2149,
1971, 1787, 1788, 1873, 2335, 2425, 2150, 3004, 2409, 1972, 2151, 1789, 1973, 2336,
2264, 2410, 1790, 3005, 2500, 3006, 2265, 2501, 1791, 1792, 2152, 2153, 2337, 1793,
2338, 1874, 2411, 1794, 2412, 1795, 2502, 2154, 2413, 1796, 2266, 2155, 1797, 1798,
1875, 2503, 2339, 1974, 2414, 1799, 1975, 2504, 2415, 1876, 2267, 2227, 2156, 2268,
1877, 2270, 3007, 2271, 1878, 1879, 1976, 2269, 1977, 2272, 2157, 2273, 2340, 3008,
1880, 3009, 1978, 1881, 1882, 2158, 1883, 2341, 1884, 2416, 1885, 2505, 2159, 1979,
1886, 2426, 2160, 3010, 1887, 2274, 2417, 1888, 1889, 1980, 3011, 1890, 3012, 1891,
1892, 2161, 1893, 2342, 2162, 2163, 2343, 2164, 2418, 1894, 2419, 1895, 2506, 2526,
2165, 2507, 2275, 2508, 1896, 2600, 3013, 1897, 2276, 2601, 1898, 1899, 1981, 1982,
2166, 1983, 2344, 2421, 1984, 2422, 2167, 2277, 2168, 2228, 2169, 1985, 2509, 1986,
2602, 2170, 3014, 1987, 2278, 2280, 3015, 1988, 2281, 1989, 1990, 3016, 1991, 1992,
2171, 1993, 2345}
Merci Jean-Marc !
Best,
Ι.
[March 6th, 2014]