Palindromaniac

(Tania Mouraud)

A371113 was recently submitted to the OEIS:

The sequence T is a succession of triples of nonnegative integers. In each triple, the digits form a palindromic pattern. The sequence starts with a(1)=0 and is always extended with the smallest available integer not yet present in the sequence.

T = 0, 1, 10, 2, 3, 32, 4, 5, 54, 6, 7, 76, 8, 9, 98, 11, 12, 111, 13, 14, 131, 15, 16, 151, 17, 18, 171, 19, 20, 291, 21, 22, 212, 23, 24, 232, 25, 26, 252, 27, 28, 272, 29, 30, 392, 31, 33, 313, 34, 35, 343, 36, 37, 363, 38, 39, 383, 40, 41, 404, 42, 43, 424, 44, 45, 444, 46, 47, 464, 48, 49, 484, 50, 51, 505, ..

Triples begin (0,1,10), (2,3,32), (4,5,54), (6,7,76), (8,9,98), ... where the palindromic pattern is clearly visible in each triple.

The above sequence T was suggested to the author by the hereunder A238880:

The sequence U is a succession of pairs of nonnegative integers. The digits, in each pair, form a palindromic pattern. The sequence starts with a(1)=0 and is always extended with the smallest available integer not yet present in the sequence.

U = 0, 10, 1, 11, 2, 12, 3, 13, 4, 14, 5, 15, 6, 16, 7, 17, 8, 18, 9, 19, 20, 102, 21, 112, 22, 122, 23, 32, 24, 42, 25, 52, 26, 62, 27, 72, 28, 82, 29, 92, 30, 103, 31, 113, 33, 133, 34, 43, 35, 53, 36, 63, 37, 73, 38, 83, 39, 93, 40, 104, 41, 114, 44, 144, 45, 54, 46, 64, 47, 74, 48, 84, 49, 94, 50, 105, 51, 115, 55, 155, 56, 65, 57, 75, 58, 85, 59, 95, 60, 106, 61, 116, 66, 166, 67, 76, 68, 86, 69, 96, 70, 107, 71, 117, 77, 177, 78, 87, 79, 97, 80, 10, ...

Example: (0,10)(1,11)(2,12)(3,13)(4,14)(5,15)... The palindromic pattern is clearly visible in each pair.

I had completely forgotten the keywords nice and look – and this gave me the idea of today's blog entry.

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Today's blog entry

The hereunder 1^st pair of terms (1,221) and the 2^nd one (12,21) share the same palindromic digit succession.

The hereunder 3^rd pair of terms (2,332) and the 4^th one (23,32) share the same palindromic digit succession.

The hereunder 5^th pair of terms (3,113) and the 6^th one (31,13) share the same palindromic digit succession. Etc.

1,221,12,21,2,332,23,32,3,113,31,13,4,114,41,14,5,115,51,15,6,116,61,16,7,117,71,17,8,118,81,18,9,119,91,19,10,2201,102,201,11,2211,112,211,20,3302,203,302,22,3322,223,322,24,142,241,42,25,152,251,52,26,162,261,62,27,172,271,72,28,182,281,82,29,192,291,92,30,1103,301,103,33,1133,331,133,...

This is, I hope, the lexicographically earliest sequence of distinct terms > 0 with this property. And a permutation of the natural numbers.

[I wonder how "nice" would look the graph of the first 10000 terms!] 

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Next day update

Giorgos Kalogeropoulos was quick to compute and extend the sequence. Two graphs were adjoined in the mail:

1,221,12,21,2,332,23,32,3,113,31,13,4,114,41,14,5,115,51,15,6,116,61,16,7,117,71,17,8,118,81,18,9,119,91,19,10,2201,102,201,11,2211,112,211,20,3302,203,302,22,3322,223,322,24,142,241,42,25,152,251,52,26,162,261,62,27,172,271,72,28,182,281,82,29,192,291,92,30,1103,301,103,33,1133,331,133,34,143,341,43,35,153,351,53,36,163,361,63,37,173,371,73,38,183,381,83,39,193,391,93,40,1104,401,104,44,1144,441,144,45,154,451,54,46,164,461,64,47,174,471,74,48,184,481,84,49,194,491,94,50,1105,501,105,55,1155,551,155,56,165,561,65,57,175,571,75,58,185,581,85,59,195,591,95,60,1106,601,106,66,1166,661,166,67,176,671,76,68,186,681,86,69,196,691,96,70,1107,701,107,77,1177,771,177,78,187,781,87,79,197,791,97,80,1108,801,108,88,1188,881,188,89,198,891,98,90,1109,901,109,99,1199,991,199,100,22001,1002,2001,101,11101,1011,1101,110,22011,1102,2011,111,22111,1112,2111,120,11021,1201,1021,121,11121,1211,1121,122,22221,1222,2221,123,1321,1231,321,124,1421,1241,421,125,1521,1251,521,126,1621,1261,621,127,1721,1271,721,128,1821,1281,821,129,1921,1291,921,130,11031,1301,1031,131,11131,1311,1131,132,2231,1322,231,134,1431,1341,431,135,1531,1351,531,136,1631,1361,631,137,1731,1371,731,138,1831,1381,831,139,1931,1391,931,140,11041,1401,1041,141,11141,1411,1141,145,1541,1451,541,146,1641,1461,641,147,1741,1471,741,148,1841,1481,841,149,1941,1491,941,150,11051,1501,1051,151,11151,1511,1151,156,1651,1561,651,157,1751,1571,751,158,1851,1581,851,159,1951,1591,951,160,11061,1601,1061,161,11161,1611,1161,167,1761,1671,761,168,1861,1681,861,169,1961,1691,961,170,11071,1701,1071,171,11171,1711,1171,178,1871,1781,871,179,1971,1791,971,180,11081,1801,1081,181,11181,1811,1181,189,1981,1891,981,190,11091,1901,1091,191,11191,1911,1191,200,33002,2003,3002,202,11202,2021,1202,204,1402,2041,402,205,1502,2051,502,206,1602,2061,602,207,1702,2071,702,208,1802,2081,802,209,1902,2091,902,210,11012,2101,1012,212,11212,2121,1212,213,1312,2131,312,214,1412,2141,412,215,1512,2151,512,216,1612,2161,612,217,1712,2171,712,218,1812,2181,812,219,1912,2191,912,220,22022,2202,2022,222,33222,2223,3222,224,1422,2241,422,225,1522,2251,522,226,1622,2261,622,227,1722,2271,722,228,1822,2281,822,229,1922,2291,922,230,11032,2301,1032,232,11232,2321,1232,233,11332,2331,1332,234,1432,2341,432,235,1532,2351,532,236,1632,2361,632,237,1732,2371,732,238,1832,2381,832,239,1932,2391,932,240,11042,2401,1042,242,11242,2421,1242,243,1342,2431,342,244,1442,2441,442,245,1542,2451,542,246,1642,2461,642,247,1742,2471,742,248,1842,2481,842,249,1942,2491,942,250,11052,2501,1052,252,11252,2521,1252,253,1352,2531,352,254,1452,2541,452,255,1552,2551,552,256,1652,2561,652,257,1752,2571,752,258,1852,2581,852,259,1952,2591,952,260,11062,2601,1062,262,11262,2621,1262,263,1362,2631,362,264,1462,2641,462,265,1562,2651,562,266,1662,2661,662,267,1762,2671,762,268,1862,2681,862,269,1962,2691,962,270,11072,2701,1072,272,11272,2721,1272,273,1372,2731,372,274,1472,2741,472,275,1572,2751,572,276,1672,2761,672,277,1772,2771,772,278,1872,2781,872,279,1972,2791,972,280,11082,2801,1082,282,11282,2821,1282,283,1382,2831,382,284,1482,2841,482,285,1582,2851,582,286,1682,2861,682,287,1782,2871,782,288,1882,2881,882,289,1982,2891,982,290,11092,2901,1092,292,11292,2921,1292,293,1392,2931,392,294,1492,2941,492,295,1592,2951,592,296,1692,2961,692,297,1792,2971,792,298,1892,2981,892,299,1992,2991,992,300,11003,3001,1003,303,11303,3031,1303,304,1403,3041,403,305,1503,3051,503,306,1603,3061,603,307,1703,3071,703,308,1803,3081,803,309,1903,3091,903,310,11013,3101,1013,311,11113,3111,1113,313,11313,3131,1313,314,1413,3141,413,315,1513,3151,513,316,1613,3161,613,317,1713,3171,713,318,1813,3181,813,319,1913,3191,913,320,11023,3201,1023,323,11323,3231,1323,324,1423,3241,423,325,1523,3251,523,326,1623,3261,623,327,1723,3271,723,328,1823,3281,823,329,1923,3291,923,330,11033,3301,1033,333,11333,3331,1333,334,1433,3341,433,335,1533,3351,533,336,1633,3361,633,337,1733,3371,733,338,1833,3381,833,339,1933,3391,933,340,11043,3401,1043,343,11343,3431,1343,344,1443,3441,443,345,1543,3451,543,346,1643,3461,643,347,1743,3471,743,348,1843,3481,843,349,1943,3491,943,350,11053,3501,1053,353,11353,3531,1353,354,1453,3541,453,355,1553,3551,553,356,1653,3561,653,357,1753,3571,753,358,1853,3581,853,359,1953,3591,953,360,11063,3601,1063,363,11363,3631,1363,364,1463,3641,463,365,1563,3651,563,366,1663,3661,663,367,1763,3671,763,368,1863,3681,863,369,1963,3691,963,370,11073,3701,1073,...

1000-term graph

2200-term graph

Jean-Marc Falcoz, a few minutes after Giorgos, confirmed the above terms. He also sent the hereunder beautiful (and fractal, as suspected) 10000-term graph.

My warm thanks to all for those computations and overwhelming artworks!

(Dall-e creation)