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hexagonal n.  and  triangular n.

First numbers that belong to both sets:

1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, 276, 325, 378, 435, 496, 561, 630, 703, 780, 861, 946, 1035, 1128, 1225, 1326, 1431, 1540, 1653, 1770, 1891, 2016, 2145, 2278, 2415, 2556, 2701, 2850, 3003, 3160, 3321, 3486, 3655, 3828, 4005, 4186, 4371, 4560, 4753, 4950, 5151, 5356, 5565, 5778, 5995, 6216, 6441, 6670, 6903, 7140, 7381, 7626, 7875, 8128, 8385, 8646, 8911, 9180, 9453, 9730, 10011, 10296, 10585, 10878, 11175, 11476, 11781, 12090, 12403, 12720, 13041, 13366, 13695, 14028, 14365, 14706, 15051, 15400, 15753, 16110, 16471, 16836, 17205, 17578, 17955, 18336, 18721, 19110, 19503, 19900, 20301, 20706, 21115, 21528, 21945, 22366, 22791, 23220, 23653, 24090, 24531, 24976, 25425, 25878, 26335, 26796, 27261, 27730, 28203, 28680, 29161, 29646, 30135, 30628, 31125, 31626, 32131, 32640, 33153, 33670, 34191, 34716, 35245, 35778, 36315, 36856, 37401, 37950, 38503, 39060, 39621, 40186, 40755, 41328, 41905, 42486, 43071, 43660, 44253, 44850, 45451, 46056, 46665, 47278, 47895, 48516, 49141, 49770, 50403, 51040, 51681, 52326, 52975, 53628, 54285, 54946, 55611, 56280, 56953, 57630, 58311, 58996, 59685, 60378, 61075, 61776, 62481, 63190, 63903, 64620, 65341, 66066, 66795, 67528, 68265, 69006, 69751, 70500, 71253, 72010, 72771, 73536, 74305, 75078, 75855, 76636, 77421, 78210, 79003, 79800, 80601, 81406, 82215, 83028, 83845, 84666, 85491, 86320, 87153, 87990, 88831, 89676, 90525, 91378, 92235, 93096, 93961, 94830, 95703, 96580, 97461, 98346, 99235, 100128, 101025, 101926, 102831, 103740, 104653, 105570, 106491, 107416, 108345, 109278, 110215, 111156, 112101, 113050, 114003, 114960, 115921, 116886, 117855, 118828, 119805, 120786, 121771, 122760, 123753, 124750, 125751, 126756, 127765, 128778, 129795, 130816, 131841, 132870, 133903, 134940, 135981, 137026, 138075, 139128, 140185, 141246, 142311, 143380, 144453, 145530, 146611, 147696, 148785, 149878, 150975, 152076, 153181, 154290, 155403, 156520.

Some larger examples :