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centered octagonal n.  and  Duffinian n.

First numbers that belong to both sets:

9, 25, 49, 81, 121, 169, 225, 289, 361, 529, 625, 729, 841, 961, 1089, 1225, 1369, 1681, 1849, 2025, 2209, 2401, 2601, 2809, 3025, 3481, 3721, 4225, 4489, 4761, 5041, 5329, 5625, 6241, 6561, 6889, 7225, 7569, 7921, 8281, 9025, 9409, 10201, 10609, 11449, 11881, 12769, 13225, 14161, 14641, 15129, 15625, 16129, 17161, 18225, 18769, 19321, 19881, 20449, 21025, 21609, 22201, 22801, 23409, 24649, 25281, 26569, 27225, 27889, 28561, 29929, 30625, 31329, 32041, 32761, 34225, 34969, 36481, 37249, 38809, 39601, 41209, 42025, 42849, 44521, 45369, 46225, 47089, 48841, 49729, 50625, 51529, 52441, 54289, 55225, 57121, 58081, 59049, 60025, 61009, 62001, 63001, 64009, 65025, 66049, 68121, 69169, 70225, 71289, 72361, 73441, 76729, 78961, 80089, 82369, 83521, 85849, 87025, 88209, 89401, 90601, 91809, 93025, 94249, 96721, 97969, 100489, 101761, 103041, 104329, 105625, 108241, 109561, 112225, 113569, 114921, 116281, 117649, 119025, 120409, 121801, 124609, 126025, 128881, 130321, 131769, 133225, 134689, 136161, 139129, 140625, 143641, 146689, 151321, 152881, 154449, 156025, 157609, 160801, 162409, 164025, 165649, 167281, 168921, 170569, 172225, 175561, 177241, 178929, 180625, 182329, 185761, 187489, 189225, 190969, 192721, 194481, 196249, 198025, 199809, 201601, 203401, 207025, 208849, 210681, 212521, 214369, 218089, 223729, 225625, 227529, 229441, 231361, 235225, 237169, 241081, 243049, 247009, 249001, 251001, 253009, 255025, 259081, 261121, 265225, 267289, 269361, 271441, 273529, 277729, 279841, 281961, 284089, 286225, 288369, 292681, 297025, 299209, 303601, 310249, 312481, 314721, 316969, 319225, 323761, 326041, 328329, 330625, 332929, 337561, 339889, 344569, 346921, 349281, 351649, 354025, 358801, 361201, 368449, 373321, 375769, 378225, 380689, 383161, 385641, 388129, 390625, 395641, 398161, 403225, 405769, 408321, 410881, 413449, 418609, 421201, 426409, 429025, 434281, 436921, 444889, 450241, 452929, 455625, 458329, 461041, 463761, 466489, 469225, 474721, 477481, 483025, 485809, 488601, 491401, 494209, 497025, 499849, 502681, 508369.

Some larger examples :