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alternating n.  and  centered triangular n.

First numbers that belong to both sets:

1, 4, 10, 85, 109, 274, 361, 694, 901, 1054, 2341, 3676, 4789, 5674, 7456, 9010, 16381, 18985, 38161, 43096, 45676, 49414, 50509, 65836, 69016, 70309, 74929, 129214, 167836, 250309, 258961, 270301, 303076, 309856, 325036, 349210, 383296, 418969, 430141, 498529, 507214, 523036, 676369, 678385, 692581, 721414, 723496, 783010, 909094, 927874, 1270981, 1290385, 1430329, 1450909, 1498501, 1696549, 2169010, 2341876, 2387074, 3212749, 3252385, 3610729, 3652741, 4101094, 4165834, 4185856, 4363096, 4549234, 4723276, 5214541, 5298781, 5434969, 6129694, 6325294, 6505210, 6549016, 7210585, 7256701, 7630921, 8167834, 8543074, 9650749, 12325234, 12523816, 14363896, 16349854, 16389496, 16909210, 18781474, 18909076, 21472309, 21654301, 23814361, 23898109, 25270381, 27278941, 27676981, 29470585, 29670385, 32329210, 34365874, 43618585, 45432769, 47452501, 47858329, 49498921, 52525210, 54387694, 54967294, 54985456, 61276909, 63618985, 69676561, 70723234, 72965476, 74141896, 76587256, 78941656, 89038981, 90547696, 90921016, 94149010, 94529674, 101438929, 101636389, 101858761, 107878321, 121054909, 123438169, 125652385, 125872141, 127858585, 129414349, 141498985, 145696321, 161476501, 163474741, 165012949, 167254561, 169830361, 181252585, 183634741, 187874509, 210189610, 210367210, 212183014, 212361454, 218545456, 230547010, 234581296, 258103210, 270567496, 276563494, 290545210, 294105010, 294567274, 294903694, 305856181, 327872161, 347678101, 363894301, 381292789, 383256361, 385898341, 389032381, 389418985, 430521634, 452549296, 456361210, 472141234, 478389034, 496341436, 498909610, 501210181, 541072585, 543296989, 543810961, 547874149, 585656761, 587614585, 589456729, 632929834, 634965076, 652365256, 670761694, 698123854, 698965474, 707670181, 709234921, 723856585, 745098985, 761232385, 781436701, 783012385, 812763010, 814789414, 818709610, 850981414, 856349014, 856707454, 905416789, 945278569, 983692909, 989232121.