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alternating n.  and  hexagonal n.

First numbers that belong to both sets:

1, 6, 45, 276, 325, 496, 561, 630, 703, 2145, 2701, 6903, 8385, 10585, 14365, 18721, 25878, 38503, 47278, 74305, 89676, 90525, 107416, 109278, 147696, 149878, 210925, 254541, 303810, 345696, 349030, 383250, 454581, 527878, 561270, 674541, 705078, 709836, 763230, 810901, 852165, 907878, 1038961, 1050525, 1212903, 1256905, 1454365, 1474903, 1830741, 2129016, 2305878, 2323090, 2383836, 2965830, 3078921, 3616705, 3812941, 4343878, 4367490, 5676765, 5812345, 6123250, 6725278, 7014385, 7210503, 7256145, 7830903, 9616305, 10321696, 14127270, 14903070, 16105650, 18141276, 18189496, 18383016, 18945090, 21238903, 21474181, 25614903, 27036981, 27450345, 29074125, 29810781, 29872585, 34109670, 34507278, 36529878, 38189430, 38927076, 43296165, 45672903, 47098365, 47214903, 58185078, 63636121, 69072381, 70323870, 76725078, 83430903, 83896581, 87456925, 89692921, 92963430, 96709278, 98329276, 98947278, 101410161, 105436981, 107890705, 125476561, 129050145, 147258541, 147670705, 163470321, 167252905, 167692141, 169896961, 187278981, 212581890, 216143236, 218729070, 236389896, 236563876, 276301278, 278185078, 301830165, 323634961, 323838525, 349470703, 361092501, 363676965, 414129810, 450705276, 450945496, 458181856, 474181410, 496109250, 496361278, 501858721, 545870361, 563052903, 583230781, 610943490, 616127856, 630107250, 638941878, 676365810, 698763036, 701232525, 785090125, 789216585, 789852385, 812347278, 838185096, 850183230, 874769878, 921470985, 961038561, 969474561, 969650703, 985036305.