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amenable n.  and  centered octagonal n.

First numbers that belong to both sets:

1, 9, 25, 49, 81, 121, 169, 225, 289, 361, 441, 529, 625, 729, 841, 961, 1089, 1225, 1369, 1521, 1681, 1849, 2025, 2209, 2401, 2601, 2809, 3025, 3249, 3481, 3721, 3969, 4225, 4489, 4761, 5041, 5329, 5625, 5929, 6241, 6561, 6889, 7225, 7569, 7921, 8281, 8649, 9025, 9409, 9801, 10201, 10609, 11025, 11449, 11881, 12321, 12769, 13225, 13689, 14161, 14641, 15129, 15625, 16129, 16641, 17161, 17689, 18225, 18769, 19321, 19881, 20449, 21025, 21609, 22201, 22801, 23409, 24025, 24649, 25281, 25921, 26569, 27225, 27889, 28561, 29241, 29929, 30625, 31329, 32041, 32761, 33489, 34225, 34969, 35721, 36481, 37249, 38025, 38809, 39601, 40401, 41209, 42025, 42849, 43681, 44521, 45369, 46225, 47089, 47961, 48841, 49729, 50625, 51529, 52441, 53361, 54289, 55225, 56169, 57121, 58081, 59049, 60025, 61009, 62001, 63001, 64009, 65025, 66049, 67081, 68121, 69169, 70225, 71289, 72361, 73441, 74529, 75625, 76729, 77841, 78961, 80089, 81225, 82369, 83521, 84681, 85849, 87025, 88209, 89401, 90601, 91809, 93025, 94249, 95481, 96721, 97969, 99225, 100489, 101761, 103041, 104329, 105625, 106929, 108241, 109561, 110889, 112225, 113569, 114921, 116281, 117649, 119025, 120409, 121801, 123201, 124609, 126025, 127449, 128881, 130321, 131769, 133225, 134689, 136161, 137641, 139129, 140625, 142129, 143641, 145161, 146689, 148225, 149769, 151321, 152881, 154449, 156025, 157609, 159201, 160801, 162409, 164025, 165649, 167281, 168921, 170569, 172225, 173889, 175561, 177241, 178929, 180625, 182329, 184041, 185761, 187489, 189225, 190969, 192721, 194481, 196249, 198025, 199809, 201601, 203401, 205209, 207025, 208849, 210681, 212521, 214369, 216225, 218089, 219961, 221841, 223729, 225625, 227529, 229441, 231361, 233289, 235225, 237169, 239121, 241081, 243049, 245025, 247009, 249001, 251001, 253009, 255025, 257049, 259081, 261121, 263169, 265225, 267289, 269361, 271441, 273529, 275625, 277729, 279841, 281961, 284089, 286225, 288369, 290521, 292681, 294849, 297025, 299209, 301401, 303601, 305809, 308025, 310249, 312481.

Some larger examples :