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alternating n.  and  pentagonal n.

First numbers that belong to both sets:

1, 5, 12, 70, 92, 145, 210, 852, 925, 2147, 2501, 3290, 3432, 3876, 4187, 4347, 6305, 6501, 12105, 12927, 25676, 27270, 27676, 45850, 58707, 69230, 69876, 70525, 76501, 78547, 98945, 101270, 123410, 161212, 270725, 349692, 367290, 567030, 707610, 709672, 743072, 814385, 832165, 870585, 909872, 1052947, 1090987, 1238967, 1830985, 1834301, 1874327, 2365676, 2503250, 2985676, 3654301, 3658985, 3696565, 4329052, 4561432, 4709090, 4725050, 4945876, 4967690, 5016947, 5696927, 5838107, 6143852, 6507292, 6563650, 7272105, 7810145, 7892507, 8121230, 8725410, 8987832, 9298905, 10901276, 12165232, 14329876, 14787830, 16165492, 16323452, 16561832, 18785012, 21852325, 23212567, 23818345, 23830301, 25090105, 27694165, 27810301, 30143692, 30305290, 30969632, 32501210, 32949610, 32963672, 34141276, 36543676, 38145252, 38327010, 38387692, 45454785, 47458125, 52141276, 54123070, 54141092, 65614187, 65852501, 67432185, 67452301, 69050945, 70503676, 72325232, 72909690, 74761870, 76187630, 83876987, 89416901, 92547610, 94561430, 101274525, 101496501, 103256165, 105458145, 109854767, 123610587, 123692301, 125030785, 145292525, 149096365, 163892587, 165054905, 169458147, 169872925, 187650745, 189838125, 210947892, 210983470, 230745412, 232385490, 232721676, 234181290, 236925652, 238549676, 238701030, 258943412, 274503412, 292985876, 294329092, 296181030, 296349676, 298165652, 307672365, 323290301, 325098565, 369256305, 383272345, 385818147, 410101070, 410547632, 416325070, 416525012, 418527072, 434307876, 436301010, 436505692, 436761612, 454705676, 476105876, 507058301, 521276925, 523292187, 527278127, 563018127, 567852545, 569078987, 569254301, 630303252, 634985650, 654703050, 692783876, 701450125, 703072925, 721836985, 729292725, 767010347, 767078187, 767892501, 810901876, 816561670, 852721052, 874387032, 890321472, 894785452, 945454301, 949676785, 963616901.