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centered square n.  and  Duffinian n.

First numbers that belong to both sets:

25, 85, 221, 265, 365, 481, 685, 841, 925, 1105, 1405, 1513, 1985, 2665, 2813, 2965, 3281, 3445, 3785, 3961, 4141, 4325, 4705, 4901, 5305, 5513, 5941, 6161, 6385, 6613, 6845, 7081, 7813, 8065, 8321, 9385, 10805, 11101, 11401, 11705, 12013, 12961, 13285, 14965, 15665, 16021, 16745, 17113, 17861, 18241, 19405, 20605, 21425, 22261, 23113, 23545, 24865, 25313, 25765, 26221, 27613, 28085, 28561, 29041, 29525, 30505, 31001, 31501, 32005, 32513, 33025, 33541, 34585, 36181, 38365, 39481, 40613, 41185, 42341, 43513, 44105, 45301, 45905, 46513, 48361, 48985, 50245, 50881, 52165, 53465, 54781, 55445, 56785, 58141, 60205, 61601, 62305, 63013, 64441, 65161, 65885, 66613, 68081, 69565, 71065, 72581, 74113, 74885, 78013, 78805, 80401, 82825, 84461, 85285, 87781, 88621, 91165, 92021, 92881, 95485, 96361, 98125, 102605, 103513, 104425, 107185, 108113, 109045, 109981, 111865, 112813, 113765, 114721, 115681, 117613, 118585, 119561, 120541, 121525, 122513, 123505, 124501, 125501, 126505, 127513, 128525, 129541, 130561, 131585, 132613, 137813, 138865, 140981, 144185, 145261, 147425, 149605, 150701, 151801, 152905, 154013, 155125, 157361, 159613, 160745, 164165, 166465, 171113, 172285, 173461, 174641, 175825, 179401, 180601, 181805, 183013, 186661, 189113, 191581, 192821, 195313, 196565, 197821, 200345, 201613, 202885, 208013, 209305, 211901, 213205, 214513, 217141, 221113, 225121, 226465, 227813, 229165, 230521, 231881, 238741, 240125, 242905, 245701, 247105, 248513, 251341, 258481, 259921, 261365, 262813, 265721, 267181, 270113, 274541, 276025, 279005, 280501, 283505, 285013, 286525, 288041, 291085, 292613, 295681, 297221, 298765, 300313, 301865, 306545, 308113, 309685, 311261, 314425, 316013, 317605, 320801, 322405, 324013, 325625, 327241, 328861, 330485, 338665, 340313, 341965, 343621, 345281, 348613, 350285, 351961, 355325, 357013, 358705, 360401, 362101, 367225, 368941, 374113, 379321, 381065, 384565, 395161, 396941, 400513, 402305, 404101, 407705, 409513, 411325, 414961, 416785, 418613, 422281, 425965, 429665, 433381, 438985, 440861, 442741, 444625.

Some larger examples :