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alternating n.  and  triangular n.

First numbers that belong to both sets:

1, 3, 6, 10, 21, 36, 45, 78, 105, 210, 276, 325, 496, 561, 630, 703, 741, 903, 1830, 2145, 2701, 5050, 6105, 6903, 8385, 9870, 10585, 12561, 14365, 18145, 18721, 23436, 25878, 29890, 30381, 32385, 36585, 38503, 38781, 41616, 47278, 50721, 67896, 70125, 74305, 81810, 83436, 85078, 89676, 90525, 96141, 107416, 109278, 121278, 123256, 125250, 147696, 149878, 163878, 210925, 214185, 230181, 232903, 254541, 258121, 290703, 301476, 303810, 323610, 345696, 349030, 383250, 414505, 454581, 490545, 498501, 527878, 545490, 549676, 561270, 634501, 674541, 705078, 709836, 725410, 763230, 810901, 852165, 858705, 874503, 907878, 941878, 1038961, 1050525, 1092981, 1098903, 1212903, 1256905, 1454365, 1474903, 1830741, 2129016, 2189278, 2305878, 2323090, 2383836, 2527876, 2545896, 2701650, 2729616, 2929410, 2965830, 3078921, 3252525, 3272961, 3616705, 3630165, 3812941, 4183278, 4323270, 4343878, 4367490, 4549636, 4769416, 5038725, 5256903, 5612925, 5676765, 5812345, 6123250, 6367096, 6525078, 6725278, 7014385, 7210503, 7214301, 7256145, 7490385, 7830903, 7850703, 7898325, 9016381, 9050385, 9616305, 9850141, 10109256, 10127250, 10321696, 10785690, 12145056, 12723490, 12743676, 12763878, 14127270, 14903070, 16105650, 16707090, 18123210, 18141276, 18189496, 18383016, 18583656, 18945090, 21238903, 21454525, 21474181, 23218705, 25052581, 25450545, 25614903, 25650703, 27014925, 27036981, 27450345, 29074125, 29810781, 29818503, 29872585, 34109670, 34349616, 34507278, 36129250, 36367656, 36529878, 36743878, 38163216, 38189430, 38927076, 43296165, 43454503, 45072765, 45672903, 47098365, 47214903, 47614161, 50125078, 50929278, 50969656, 58185078, 58541610, 61012581, 61410903, 61810521, 63636121, 63692541, 65054121, 69072381, 69696721, 70323870, 72727830, 74963890, 76725078, 76985436, 83030941, 83430903, 83896581, 85418985, 87456925, 89652745, 89692921, 90969816, 92541210, 92963430, 96389670, 96709278, 96945850, 98329276, 98947278, 101410161, 105436981, 105858525, 107890705, 125476561, 129050145, 141834903, 145018965, 147258541, 147670705, 163470321, 167252905, 167692141, 169896961, 183830725, 187278981, 189238785, 210709656, 212581890, 216143236, 218729070, 236389896, 236563876, 238765878, 256567878, 258383278, 270525430, 272109456, 272389470, 276301278, 276701050, 276983416, 278185078, 294965616, 301830165, 323634961, 323838525, 329898141, 347490703, 349470703, 361092501, 363676965, 414129810, 434785816, 450705276, 450945496, 458181856, 470769270, 474181410, 474581836, 474705078, 476185230.

Some larger examples :