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cyclic n.  and  decagonal n.

First numbers that belong to both sets:

1, 85, 451, 1105, 1387, 2047, 3277, 4795, 5365, 6601, 8695, 11077, 13747, 14701, 16705, 17755, 19951, 24727, 27307, 28645, 31417, 35815, 40501, 42127, 47197, 50737, 52555, 68251, 74665, 76867, 81367, 83665, 88357, 90751, 103201, 105787, 111055, 127627, 130501, 145351, 148417, 154645, 157807, 184255, 194701, 209077, 216457, 220195, 227767, 243295, 251251, 267547, 275887, 288637, 306085, 315001, 319507, 342517, 356707, 385951, 390937, 416347, 431977, 437251, 453265, 464101, 480595, 486157, 497377, 514447, 537655, 549451, 555397, 573427, 585607, 604117, 610351, 622915, 629245, 642001, 648427, 667897, 681037, 687655, 741751, 748657, 762565, 769567, 783667, 790765, 805057, 812251, 848701, 870955, 878437, 893497, 916327, 924001, 939445, 947215, 962851, 970717, 994507, 1034797, 1084201, 1092547, 1134757, 1143295, 1169101, 1195195, 1212751, 1266187, 1293337, 1302451, 1376515, 1385917, 1404817, 1414315, 1433407, 1491451, 1501237, 1520905, 1530787, 1550647, 1580677, 1590751, 1610995, 1641601, 1651867, 1682857, 1703677, 1735147, 1745701, 1766905, 1798951, 1809697, 1842127, 1863907, 1907851, 1930015, 1963501, 2008597, 2111935, 2135251, 2146957, 2182267, 2205967, 2217865, 2241757, 2253751, 2314201, 2326387, 2350855, 2363137, 2387797, 2425027, 2437501, 2462545, 2475115, 2513017, 2576827, 2589685, 2615497, 2654455, 2667505, 2693701, 2746477, 2813167, 2894251, 2907877, 2948947, 2976487, 2990305, 3102001, 3116107, 3144415, 3187117, 3230107, 3375487, 3449377, 3509065, 3539101, 3554167, 3584395, 3599557, 3629977, 3675847, 3737455, 3768451, 3783997, 3862207, 3909517, 3925351, 3973045, 4005001, 4021027, 4069297, 4101637, 4150387, 4166701, 4248751, 4265257, 4314967, 4398457, 4415251, 4499701, 4516687, 4567837, 4602097, 4653727, 4775317, 4810345, 4827907, 4863127, 4916197, 4933951, 4969555, 4987405, 5023201, 5095177, 5131357, 5149495, 5185867, 5204101, 5240665, 5295751, 5369647, 5406787, 5425405, 5462737, 5481451, 5575501, 5632315, 5689417, 5746807, 5766001, 5862451, 5881837, 5920705, 5979247, 6057751, 6097195, 6116965, 6176467, 6216295, 6236257, 6276277, 6336547, 6457951, 6478297, 6539527, 6580507, 6642217, 6662851, 6787327, 6849997, 6891937, 6955087, 6976201, 7232065, 7275157, 7405201, 7426987, 7470655, 7536397, 7624501, 7690915, 7735351, 7757617, 7824607, 7869427, 7936897, 7959451, 8004655, 8095447, 8141035, 8163877, 8209657, 8278567, 8301601.

Some larger examples :