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amenable n.  and  centered triangular n.

First numbers that belong to both sets:

1, 64, 85, 109, 136, 316, 361, 409, 460, 760, 829, 901, 976, 1396, 1489, 1585, 1684, 2224, 2341, 2461, 2584, 3244, 3385, 3529, 3676, 4456, 4621, 4789, 4960, 5860, 6049, 6241, 6436, 7456, 7669, 7885, 8104, 9244, 9481, 9721, 9964, 11224, 11485, 11749, 12016, 13396, 13681, 13969, 14260, 15760, 16069, 16381, 16696, 18316, 18649, 18985, 19324, 21064, 21421, 21781, 22144, 24004, 24385, 24769, 25156, 27136, 27541, 27949, 28360, 30460, 30889, 31321, 31756, 33976, 34429, 34885, 35344, 37684, 38161, 38641, 39124, 41584, 42085, 42589, 43096, 45676, 46201, 46729, 47260, 49960, 50509, 51061, 51616, 54436, 55009, 55585, 56164, 59104, 59701, 60301, 60904, 63964, 64585, 65209, 65836, 69016, 69661, 70309, 70960, 74260, 74929, 75601, 76276, 79696, 80389, 81085, 81784, 85324, 86041, 86761, 87484, 91144, 91885, 92629, 93376, 97156, 97921, 98689, 99460, 103360, 104149, 104941, 105736, 109756, 110569, 111385, 112204, 116344, 117181, 118021, 118864, 123124, 123985, 124849, 125716, 130096, 130981, 131869, 132760, 137260, 138169, 139081, 139996, 144616, 145549, 146485, 147424, 152164, 153121, 154081, 155044, 159904, 160885, 161869, 162856, 167836, 168841, 169849, 170860, 175960, 176989, 178021, 179056, 184276, 185329, 186385, 187444, 192784, 193861, 194941, 196024, 201484, 202585, 203689, 204796, 210376, 211501, 212629, 213760, 219460, 220609, 221761, 222916, 228736, 229909, 231085, 232264, 238204, 239401, 240601, 241804, 247864, 249085, 250309, 251536, 257716, 258961, 260209, 261460, 267760, 269029, 270301, 271576, 277996, 279289, 280585, 281884, 288424, 289741, 291061, 292384, 299044, 300385, 301729, 303076, 309856, 311221, 312589, 313960, 320860, 322249, 323641, 325036, 332056, 333469, 334885, 336304, 343444, 344881, 346321, 347764, 355024, 356485, 357949, 359416, 366796, 368281, 369769, 371260, 378760, 380269, 381781, 383296, 390916, 392449, 393985, 395524, 403264, 404821, 406381, 407944, 415804, 417385, 418969, 420556, 428536, 430141, 431749, 433360, 441460, 443089, 444721, 446356, 454576, 456229, 457885, 459544, 467884, 469561, 471241.

Some larger examples :