True so far
Question (by me, Eric Angelini) was (minor typos deleted) :
> Hello Math-Fun
and Seqfan,
>
> I've just sent this to the OEIS :
10 12 13 14 15 16 17 18 19 20 23 24 25 26 27 28 29 30
34 35 36 37 38 39 40 45 46 47 48 49 50 56 57 58 59 60
67 68 69 70 78 79 80 89 90 102 103 104 105 106 107
108 109 112 113 114 115 116 117 118 119 123 124 125
126 127 128 129 134 135 136 137 138 139 145 146 147
148 149 156 157 158 159 167 168 169 178 179 180 189...
> Description :
"True so far"-sequence. Last digit of a(n) must be seen
as a glyph and preceding digits as a quantity. So
"10"
reads [one "0"] and "12" [one
"2"] -- which are both true
statements: there is only one "0" glyph so far in
the
sequence when [10] is read, and there is only one
"2"
glyph when [12] is read. The sequence is built with
[a(n+1)-a(n)] being minimal and
a(n+1) always "true so
far". This explains why integers [11], [21], [22],
[31],
etc. are not in: their statements
are false.
The nice substring ...1112,1113,1114,1115,1116,1117...
appears in the sequence -- which
means that so far the
whole sequence has used 111 "2", 111
"3", 111 "4", 111 "5",
111 "6" and 111 "7"...
> Question which ruined my sleep tonight:
>
> Will the sequence ever stop?
>
> ... my intuition says yes...
> ... could someone compute this and check for some more integers?
>
> Thanks,
> É.
« True So far »
2024 terms computed by Chuck Seggelin,
Feb. 22nd, 2005.
The sequence is
finite.
10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 23, 24, 25,
26, 27, 28, 29, 30, 34, 35, 36, 37, 38, 39, 40, 45, 46, 47, 48, 49, 50, 56, 57,
58, 59, 60, 67, 68, 69, 70, 78, 79, 80, 89, 90, 102, 103, 104, 105, 106, 107,
108, 109, 112, 113, 114, 115, 116, 117, 118, 119, 123, 124, 125, 126, 127, 128,
129, 134, 135, 136, 137, 138, 139, 145, 146, 147, 148, 149, 156, 157, 158, 159,
167, 168, 169, 178, 179, 180, 189, 190, 192, 193, 194, 195, 196, 197, 203, 204,
205, 206, 207, 208, 213, 214, 215, 216, 217, 218, 223, 224, 225, 226, 227, 228,
234, 235, 236, 237, 238, 245, 246, 247, 248, 256, 257, 258, 260, 267, 268, 269,
270, 278, 279, 280, 283, 284, 285, 289, 290, 293, 294, 295, 304, 305, 306, 307,
314, 315, 316, 317, 324, 325, 326, 327, 334, 335, 336, 337, 338, 339, 340, 345,
346, 347, 348, 349, 350, 356, 357, 358, 359, 360, 367, 368, 369, 370, 378, 379,
380, 389, 390, 405, 406, 407, 408, 409, 415, 416, 417, 418, 419, 425, 426, 427,
428, 429, 435, 436, 437, 438, 439, 445, 446, 447, 448, 449, 450, 456, 457, 458,
459, 460, 467, 468, 469, 470, 478, 479, 480, 489, 490, 506, 507, 508, 509, 516,
517, 518, 519, 526, 527, 528, 529, 536, 537, 538, 539, 540, 546, 547, 548, 549,
550, 556, 557, 558, 559, 560, 567, 568, 569, 570, 578, 579, 580, 589, 590, 607,
608, 609, 617, 618, 619, 627, 628, 629, 630, 637, 638, 639, 640, 647, 648, 649,
650, 657, 658, 659, 660, 667, 668, 669, 670, 678, 679, 680, 689, 690, 708, 709,
718, 719, 720, 728, 729, 730, 738, 739, 740, 748, 749, 750, 758, 759, 760, 768,
769, 770, 778, 779, 780, 789, 790, 809, 810, 819, 820, 829, 830, 839, 840, 849,
850, 859, 860, 869, 870, 879, 880, 889, 890, 919, 932, 942, 943, 952, 953, 962,
963, 971, 972, 973, 981, 982, 983, 991, 992, 993, 1002, 1003, 1012, 1013, 1014,
1023, 1024, 1025, 1026, 1034, 1035, 1036, 1037, 1038, 1045, 1046, 1047, 1048,
1056, 1057, 1058, 1062, 1067, 1068, 1072, 1078, 1082, 1083, 1084, 1085, 1092,
1093, 1094, 1095, 1096, 1097, 1102, 1103, 1104, 1105, 1106, 1107, 1112, 1113,
1114, 1115, 1116, 1117, 1123, 1124, 1125, 1126, 1127, 1128, 1134, 1135, 1136,
1137, 1138, 1145, 1146, 1147, 1148, 1156, 1157, 1158, 1167, 1168, 1178, 1179,
1182, 1183, 1184, 1185, 1186, 1189, 1192, 1193, 1194, 1195, 1196, 1197, 1203,
1204, 1205, 1206, 1207, 1213, 1214, 1215, 1216, 1217, 1223, 1224, 1225, 1226,
1227, 1234, 1235, 1236, 1237, 1245, 1246, 1247, 1248, 1256, 1257, 1258, 1259,
1267, 1268, 1269, 1273, 1278, 1279, 1283, 1284, 1289, 1293, 1294, 1295, 1296,
1304, 1305, 1306, 1307, 1314, 1315, 1316, 1317, 1318, 1324, 1325, 1326, 1327,
1328, 1334, 1335, 1336, 1337, 1338, 1339, 1345, 1346, 1347, 1348, 1349, 1356,
1357, 1358, 1359, 1367, 1368, 1369, 1378, 1379, 1389, 1394, 1395, 1396, 1397,
1398, 1405, 1406, 1407, 1408, 1415, 1416, 1417, 1418, 1425, 1426, 1427, 1428,
1435, 1436, 1437, 1438, 1445, 1446, 1447, 1448, 1449, 1456, 1457, 1458, 1459,
1467, 1468, 1469, 1470, 1478, 1479, 1480, 1489, 1490, 1495, 1496, 1506, 1507,
1508, 1516, 1517, 1518, 1526, 1527, 1528, 1529, 1530, 1536, 1537, 1538, 1539,
1540, 1546, 1547, 1548, 1549, 1550, 1556, 1557, 1558, 1559, 1560, 1567, 1568,
1569, 1570, 1578, 1579, 1580, 1589, 1590, 1607, 1608, 1609, 1617, 1618, 1619,
1627, 1628, 1629, 1630, 1637, 1638, 1639, 1640, 1647, 1648, 1649, 1650, 1657,
1658, 1659, 1660, 1667, 1668, 1669, 1670, 1678, 1679, 1680, 1689, 1690, 1708,
1709, 1718, 1719, 1720, 1728, 1729, 1730, 1738, 1739, 1740, 1748, 1749, 1750,
1758, 1759, 1760, 1768, 1769, 1770, 1778, 1779, 1780, 1789, 1790, 1809, 1810,
1819, 1820, 1829, 1830, 1839, 1840, 1849, 1850, 1859, 1860, 1862, 1869, 1870,
1872, 1879, 1880, 1882, 1889, 1890, 1892, 1902, 1910, 1912, 1920, 1930, 1932,
1940, 1942, 1950, 1952, 1960, 1962, 1963, 1970, 1972, 1973, 1974, 1980, 1982,
1983, 1984, 1990, 1992, 1993, 1994, 2003, 2004, 2013, 2014, 2015, 2023, 2024,
2025, 2034, 2035, 2045, 2046, 2053, 2056, 2057, 2063, 2064, 2067, 2073, 2074,
2083, 2084, 2085, 2093, 2094, 2095, 2096, 2097, 2103, 2104, 2105, 2106, 2107,
2113, 2114, 2115, 2116, 2117, 2123, 2124, 2125, 2126, 2127, 2128, 2134, 2135,
2136, 2137, 2138, 2145, 2146, 2147, 2148, 2156, 2157, 2158, 2167, 2168, 2178,
2183, 2184, 2185, 2186, 2187, 2193, 2194, 2195, 2196, 2197, 2203, 2204, 2205,
2206, 2207, 2213, 2214, 2215, 2216, 2217, 2223, 2224, 2225, 2226, 2227, 2234,
2235, 2236, 2237, 2238, 2245, 2246, 2247, 2248, 2256, 2257, 2258, 2267, 2268,
2278, 2283, 2284, 2285, 2286, 2287, 2293, 2294, 2295, 2296, 2297, 2304, 2305,
2306, 2307, 2314, 2315, 2316, 2317, 2324, 2325, 2326, 2327, 2334, 2335, 2336,
2337, 2338, 2345, 2346, 2347, 2348, 2356, 2357, 2358, 2359, 2367, 2368, 2369,
2378, 2379, 2384, 2389, 2394, 2395, 2396, 2397, 2405, 2406, 2407, 2408, 2415,
2416, 2417, 2418, 2425, 2426, 2427, 2428, 2435, 2436, 2437, 2438, 2439, 2445,
2446, 2447, 2448, 2449, 2456, 2457, 2458, 2459, 2467, 2468, 2469, 2478, 2479,
2480, 2489, 2490, 2495, 2496, 2497, 2506, 2507, 2508, 2516, 2517, 2518, 2526,
2527, 2528, 2530, 2536, 2537, 2538, 2539, 2540, 2546, 2547, 2548, 2549, 2550,
2556, 2557, 2558, 2559, 2560, 2567, 2568, 2569, 2570, 2578, 2579, 2580, 2589,
2590, 2607, 2608, 2609, 2617, 2618, 2619, 2627, 2628, 2629, 2630, 2637, 2638,
2639, 2640, 2647, 2648, 2649, 2650, 2657, 2658, 2659, 2660, 2667, 2668, 2669,
2670, 2678, 2679, 2680, 2689, 2690, 2708, 2709, 2718, 2719, 2720, 2728, 2729,
2730, 2738, 2739, 2740, 2748, 2749, 2750, 2758, 2759, 2760, 2768, 2769, 2770,
2778, 2779, 2780, 2789, 2790, 2809, 2810, 2819, 2820, 2829, 2830, 2839, 2840,
2849, 2850, 2859, 2860, 2869, 2870, 2879, 2880, 2889, 2890, 2903, 2910, 2913,
2920, 2923, 2930, 2940, 2943, 2950, 2953, 2960, 2963, 2970, 2973, 2980, 2983,
2984, 2990, 2993, 2994, 3004, 3005, 3014, 3015, 3024, 3025, 3026, 3027, 3034,
3035, 3036, 3037, 3038, 3045, 3046, 3047, 3048, 3056, 3057, 3058, 3067, 3068,
3078, 3084, 3085, 3086, 3087, 3094, 3095, 3096, 3097, 3104, 3105, 3106, 3107,
3114, 3115, 3116, 3117, 3124, 3125, 3126, 3127, 3128, 3134, 3135, 3136, 3137,
3138, 3145, 3146, 3147, 3148, 3156, 3157, 3158, 3167, 3168, 3178, 3179, 3184,
3185, 3186, 3189, 3194, 3195, 3196, 3197, 3204, 3205, 3206, 3207, 3214, 3215,
3216, 3217, 3224, 3225, 3226, 3227, 3228, 3234, 3235, 3236, 3237, 3238, 3239,
3245, 3246, 3247, 3248, 3249, 3256, 3257, 3258, 3259, 3267, 3268, 3269, 3278,
3279, 3289, 3294, 3295, 3296, 3297, 3298, 3304, 3305, 3306, 3307, 3308, 3314,
3315, 3316, 3317, 3318, 3324, 3325, 3326, 3327, 3328, 3334, 3335, 3336, 3337,
3338, 3345, 3346, 3347, 3348, 3349, 3356, 3357, 3358, 3359, 3367, 3368, 3369,
3378, 3379, 3389, 3394, 3395, 3396, 3397, 3398, 3405, 3406, 3407, 3408, 3415,
3416, 3417, 3418, 3425, 3426, 3427, 3428, 3435, 3436, 3437, 3438, 3445, 3446,
3447, 3448, 3449, 3456, 3457, 3458, 3459, 3467, 3468, 3469, 3478, 3479, 3489,
3495, 3496, 3497, 3498, 3506, 3507, 3508, 3516, 3517, 3518, 3526, 3527, 3528,
3530, 3536, 3537, 3538, 3539, 3540, 3546, 3547, 3548, 3549, 3550, 3556, 3557,
3558, 3559, 3560, 3567, 3568, 3569, 3570, 3578, 3579, 3580, 3589, 3590, 3607,
3608, 3609, 3617, 3618, 3619, 3627, 3628, 3629, 3630, 3637, 3638, 3639, 3640,
3647, 3648, 3649, 3650, 3657, 3658, 3659, 3660, 3667, 3668, 3669, 3670, 3678,
3679, 3680, 3689, 3690, 3708, 3709, 3718, 3719, 3720, 3728, 3729, 3730, 3738,
3739, 3740, 3748, 3749, 3750, 3758, 3759, 3760, 3768, 3769, 3770, 3778, 3779,
3780, 3789, 3790, 3809, 3810, 3819, 3820, 3829, 3830, 3839, 3840, 3849, 3850,
3859, 3860, 3869, 3870, 3879, 3880, 3889, 3890, 3919, 3939, 3959, 3979, 3995,
4005, 4006, 4008, 4009, 4015, 4016, 4017, 4018, 4019, 4025, 4026, 4027, 4028,
4029, 4035, 4036, 4037, 4038, 4039, 4045, 4046, 4047, 4048, 4049, 4056, 4057,
4058, 4059, 4067, 4068, 4069, 4078, 4079, 4089, 4095, 4096, 4097, 4098, 4105,
4106, 4107, 4108, 4115, 4116, 4117, 4118, 4125, 4126, 4127, 4128, 4135, 4136,
4137, 4138, 4139, 4145, 4146, 4147, 4148, 4149, 4156, 4157, 4158, 4159, 4167,
4168, 4169, 4178, 4179, 4189, 4195, 4196, 4197, 4198, 4205, 4206, 4207, 4208,
4215, 4216, 4217, 4218, 4225, 4226, 4227, 4228, 4235, 4236, 4237, 4238, 4239,
4245, 4246, 4247, 4248, 4249, 4256, 4257, 4258, 4259, 4267, 4268, 4269, 4278,
4279, 4289, 4295, 4296, 4297, 4298, 4305, 4306, 4307, 4308, 4315, 4316, 4317,
4318, 4325, 4326, 4327, 4328, 4335, 4336, 4337, 4338, 4339, 4345, 4346, 4347,
4348, 4349, 4356, 4357, 4358, 4359, 4367, 4368, 4369, 4378, 4379, 4389, 4395,
4396, 4397, 4398, 4405, 4406, 4407, 4408, 4415, 4416, 4417, 4418, 4425, 4426,
4427, 4428, 4435, 4436, 4437, 4438, 4439, 4445, 4446, 4447, 4448, 4449, 4456,
4457, 4458, 4459, 4467, 4468, 4469, 4478, 4479, 4480, 4489, 4490, 4495, 4496,
4497, 4506, 4507, 4508, 4516, 4517, 4518, 4526, 4527, 4528, 4530, 4536, 4537,
4538, 4539, 4540, 4546, 4547, 4548, 4549, 4550, 4556, 4557, 4558, 4559, 4560, 4567,
4568, 4569, 4570, 4578, 4579, 4580, 4589, 4590, 4607, 4608, 4609, 4617, 4618,
4619, 4627, 4628, 4629, 4630, 4637, 4638, 4639, 4640, 4647, 4648, 4649, 4650,
4657, 4658, 4659, 4660, 4667, 4668, 4669, 4670, 4678, 4679, 4680, 4689, 4690,
4708, 4709, 4718, 4719, 4720, 4728, 4729, 4730, 4738, 4739, 4740, 4748, 4749,
4750, 4758, 4759, 4760, 4768, 4769, 4770, 4778, 4779, 4780, 4789, 4790, 4809,
4810, 4819, 4820, 4829, 4830, 4839, 4840, 4849, 4850, 4859, 4860, 4869, 4870,
4879, 4880, 4889, 4890, 4919, 4939, 4959, 4979, 4995, 5006, 5008, 5009, 5016,
5017, 5018, 5019, 5026, 5027, 5028, 5029, 5036, 5037, 5038, 5039, 5046, 5047,
5048, 5049, 5056, 5057, 5058, 5059, 5067, 5068, 5069, 5078, 5079, 5089, 5096,
5097, 5098, 5106, 5107, 5108, 5116, 5117, 5118, 5126, 5127, 5128, 5129, 5136,
5137, 5138, 5139, 5146, 5147, 5148, 5149, 5156, 5157, 5158, 5159, 5167, 5168,
5169, 5178, 5179, 5189, 5196, 5197, 5198, 5206, 5207, 5208, 5216, 5217, 5218,
5226, 5227, 5228, 5229, 5236, 5237, 5238, 5239, 5246, 5247, 5248, 5249, 5256,
5257, 5258, 5259, 5267, 5268, 5269, 5278, 5279, 5289, 5296, 5297, 5298, 5306,
5307, 5308, 5316, 5317, 5318, 5326, 5327, 5328, 5329, 5336, 5337, 5338, 5339,
5340, 5346, 5347, 5348, 5349, 5350, 5356, 5357, 5358, 5359, 5360, 5367, 5368,
5369, 5370, 5378, 5379, 5380, 5389, 5390, 5406, 5407, 5408, 5409, 5416, 5417,
5418, 5419, 5426, 5427, 5428, 5429, 5436, 5437, 5438, 5439, 5440, 5446, 5447,
5448, 5449, 5450, 5456, 5457, 5458, 5459, 5460, 5467, 5468, 5469, 5470, 5478,
5479, 5480, 5489, 5490, 5506, 5507, 5508, 5509, 5516, 5517, 5518, 5519, 5526,
5527, 5528, 5529, 5536, 5537, 5538, 5539, 5540, 5546, 5547, 5548, 5549, 5550,
5556, 5557, 5558, 5559, 5560, 5567, 5568, 5569, 5570, 5578, 5579, 5580, 5589,
5590, 5607, 5608, 5609, 5617, 5618, 5619, 5627, 5628, 5629, 5630, 5637, 5638,
5639, 5640, 5647, 5648, 5649, 5650, 5657, 5658, 5659, 5660, 5667, 5668, 5669,
5670, 5678, 5679, 5680, 5689, 5690, 5708, 5709, 5718, 5719, 5720, 5728, 5729,
5730, 5738, 5739, 5740, 5748, 5749, 5750, 5758, 5759, 5760, 5768, 5769, 5770,
5778, 5779, 5780, 5789, 5790, 5809, 5810, 5819, 5820, 5829, 5830, 5839, 5840,
5849, 5850, 5859, 5860, 5869, 5870, 5879, 5880, 5889, 5890, 5919, 5939, 5959,
5979, 6008, 6017, 6018, 6027, 6028, 6037, 6038, 6047, 6048, 6057, 6058, 6067,
6068, 6078, 6087, 6097, 6098, 6107, 6108, 6117, 6118, 6127, 6128, 6137, 6138,
6147, 6148, 6157, 6158, 6167, 6168, 6178, 6187, 6197, 6198, 6207, 6208, 6217,
6218, 6227, 6228, 6237, 6238, 6247, 6248, 6257, 6258, 6267, 6268, 6278, 6287,
6297, 6298, 6307, 6308, 6317, 6318, 6327, 6328, 6337, 6338, 6347, 6348, 6357,
6358, 6367, 6368, 6378, 6387, 6397, 6398, 6407, 6408, 6417, 6418, 6427, 6428,
6437, 6438, 6447, 6448, 6457, 6458, 6467, 6468, 6478, 6487, 6497, 6498, 6507,
6508, 6517, 6518, 6527, 6528, 6537, 6538, 6547, 6548, 6557, 6558, 6567, 6568,
6578, 6587, 6597, 6598, 6607, 6608, 6617, 6618, 6627, 6628, 6637, 6638, 6647,
6648, 6657, 6658, 6667, 6668, 6678, 6687, 6697, 6698, 6708, 6718, 6728, 6738,
6748, 6758, 6768, 6778, 7676, 7696, 7706, 7716, 7726, 7736, 7746, 7756, 8201,
8252, 8272, 8292, 8302, 8312, 8553, 8563, 8573, 8583, 8593, 8603, 8613, 8623,
8724, 8734, 8925, 8935, 8945
(last term of the sequence).
Final values for each digit:
0: 624
1: 822
2: 834
3: 864
4: 874
5: 894
6: 779
7: 697
8: 697
9: 617
Eric’s
Format:
10 12
13 14 15
16 17 18
19
20 23 24
25 26 27
28 29
30 34 35
36 37 38
39
40 45
46 47 48
49
50
56 57 58
59
60
67 68 69
70
78 79
80
89
90
102 103 104
105 106 107
108 109
112 113 114
115 116 117
118 119
123 124
125 126 127
128 129
134 135
136 137 138
139
145 146 147
148 149
156 157
158 159
167 168 169
178 179
180 189
190 192 193
194 195 196
197
203 204 205
206 207 208
213 214 215
216 217 218
223 224 225
226 227 228
234 235
236 237 238
245 246 247
248
256 257 258
260
267 268 269
270
278 279
280 283 284
285
289
290 293
294 295
304 305
306 307
314 315
316 317
324 325
326 327
334 335
336 337
338 339
340 345 346
347 348 349
350
356 357 358
359
360
367 368 369
370
378 379
380
389
390
405 406 407
408 409
415 416
417 418 419
425 426 427
428 429
435 436 437
438 439
445 446
447 448 449
450
456 457 458
459
460
467 468 469
470 478 479
480
489
490
506 507 508
509
516 517 518
519
526 527 528
529
536 537 538
539
540
546 547 548
549
550 556 557
558 559
560
567 568 569
570
578 579
580 589
590
607 608 609
617 618 619
627 628
629
630
637 638 639
640
647 648 649
650
657 658 659
660
667 668 669
670
678 679
680
689
690
708 709
718 719
720
728 729
730 738 739
740
748 749
750
758 759
760 768 769
770
778 779
780
789
790
809
810
819
820
829
830
839
840 849
850
859
860
869
870 879
880
889
890
919
932
942 943
952 953
962 963
971 972
973
981 982
983
991 992
993
1002 1003
1012 1013 1014
1023 1024 1025
1026
1034 1035
1036 1037 1038
1045 1046 1047
1048
1056 1057 1058
1062 1067 1068
1072
1078
1082 1083 1084
1085
1092 1093 1094
1095 1096 1097
1102 1103 1104
1105 1106 1107
1112 1113 1114
1115 1116 1117
1123 1124 1125
1126 1127 1128
1134 1135
1136 1137 1138
1145 1146
1147 1148
1156 1157 1158
1167 1168
1178 1179
1182 1183 1184
1185 1186 1189
1192 1193 1194
1195 1196 1197
1203 1204 1205
1206 1207
1213 1214
1215 1216 1217
1223 1224 1225
1226 1227
1234 1235
1236 1237
1245 1246 1247
1248
1256 1257
1258 1259
1267 1268 1269
1273
1278 1279
1283 1284 1289
1293 1294 1295
1296
1304 1305
1306 1307
1314 1315
1316 1317 1318
1324
1325 1326 1327
1328
1334 1335
1336 1337 1338
1339
1345 1346 1347
1348 1349
1356 1357 1358
1359
1367 1368 1369
1378 1379
1389
1394 1395
1396 1397 1398
1405 1406 1407
1408
1415 1416 1417
1418
1425
1426 1427 1428
1435 1436 1437
1438
1445 1446 1447
1448 1449
1456 1457 1458
1459
1467 1468 1469
1470
1478 1479
1480
1489
1490 1495 1496
1506 1507 1508
1516 1517 1518
1526 1527 1528
1529
1530
1536 1537 1538
1539
1540
1546 1547 1548
1549
1550
1556 1557 1558
1559
1560 1567 1568
1569
1570
1578 1579
1580
1589
1590
1607 1608
1609
1617 1618 1619
1627 1628 1629
1630 1637 1638
1639
1640
1647 1648 1649
1650
1657 1658 1659
1660
1667 1668 1669
1670
1678 1679
1680
1689
1690
1708 1709
1718 1719
1720
1728 1729
1730
1738 1739
1740 1748 1749
1750
1758 1759
1760
1768 1769
1770 1778 1779
1780
1789
1790
1809
1810 1819
1820
1829
1830
1839
1840
1849
1850
1859
1860 1862
1869
1870 1872
1879
1880 1882
1889
1890 1892
1902
1910 1912
1920
1930 1932
1940 1942
1950 1952
1960 1962 1963
1970 1972 1973
1974
1980 1982 1983
1984
1990 1992 1993
1994
2003 2004
2013 2014 2015
2023 2024 2025
2034 2035
2045 2046
2053 2056 2057