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centered triangular n.  and  happy n.

First numbers that belong to both sets:

1, 10, 19, 31, 109, 409, 694, 901, 2224, 2584, 3676, 4960, 6049, 6436, 6634, 7039, 9010, 12286, 12835, 13114, 13969, 16696, 17659, 19666, 25156, 42085, 42589, 43606, 49414, 60904, 69661, 70309, 71614, 88210, 90406, 94879, 95635, 97156, 98689, 101791, 109756, 116344, 119710, 127459, 131869, 138169, 142759, 149311, 151210, 160885, 161869, 175960, 179056, 188506, 190639, 191710, 193861, 197110, 208135, 209254, 211501, 212629, 222916, 231085, 238204, 249085, 251536, 265231, 275419, 297706, 303076, 316711, 329239, 336304, 343444, 350659, 355024, 381781, 383296, 389386, 390916, 398611, 433360, 451279, 454576, 524809, 559066, 564574, 571960, 581260, 590635, 596296, 605791, 613441, 688510, 713116, 715186, 729760, 778681, 815860, 840379, 856171, 885889, 888196, 911431, 913771, 920809, 927874, 937336, 939709, 942085, 949231, 1000009, 1007371, 1014760, 1019701, 1052110, 1064710, 1085026, 1092694, 1134046, 1186816, 1189486, 1208260, 1235335, 1251724, 1335289, 1369471, 1378084, 1380961, 1389610, 1483546, 1498501, 1537735, 1608391, 1614610, 1649029, 1683811, 1722169, 1780516, 1793614, 1849816, 1920439, 1923835, 1964821, 1982026, 1988929, 2055106, 2062135, 2079760, 2093914, 2122366, 2151010, 2172619, 2227114, 2260135, 2271196, 2300824, 2311984, 2345626, 2353135, 2364421, 2436526, 2471059, 2509714, 2548669, 2568259, 2580049, 2623510, 2627479, 2635426, 2707489, 2788699, 2825635, 2891899, 2908585, 2988499, 3060919, 3099610, 3261226, 3270079, 3278944, 3310066, 3395281, 3417886, 3422416, 3467881, 3509056, 3518239, 3573589, 3671491, 3770716, 3799309, 3852010, 3856819, 3914761, 3939031, 3992689, 4130911, 4150849, 4165834, 4317169, 4388716, 4481569, 4549234, 4723276, 4744594, 4851904, 4878919, 4889746, 4895164, 4993201, 5009635, 5059099, 5108806, 5142079, 5220136, 5231335, 5355316, 5372335, 5446396, 5469286, 5549779, 5677456, 5724244, 5735971, 5794786, 5806585, 5818396, 5847976, 6009004, 6141829, 6172219, 6214891, 6233224, 6263839, 6505210, 6530224, 6586681, 6599260, 6713011, 6859774, 6891889, 6949885, 6956344, 6982210, 7001641, 7014610, 7047085, 7066606, 7073119, 7158061, 7223746, 7429276, 7476085, 7482784, 7489486, 7509610, 7624156, 7658011, 7780510, 7862716, 7876459, 8014549, 8056210, 8084044, 8104951, 8111926, 8139856, 8167834, 8174836, 8216911, 8280226, 8400484, 8614819, 8622010, 8658010, 8817301, 8860996, 8890186.

Some larger examples :