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centered square n.  and  congruent n.

First numbers that belong to both sets:

5, 13, 41, 61, 85, 145, 181, 221, 265, 313, 365, 421, 613, 685, 761, 925, 1013, 1105, 1201, 1301, 1405, 1513, 1625, 1741, 1861, 2113, 2245, 2381, 2813, 2965, 3281, 3445, 3613, 4141, 4325, 4705, 4901, 5101, 5725, 5941, 6613, 6845, 7321, 7565, 7813, 8581, 8845, 9661, 9941, 10805, 11101, 11401, 12013, 12325, 13285, 13613, 14281, 14621, 14965, 15665, 16021, 16381, 17113, 17485, 17861, 19013, 19405, 20201, 20605, 21013, 22261, 22685, 23981, 24421, 25313, 25765, 26221, 26681, 27613, 28085, 29041, 29525, 30013, 31001, 31501, 32005, 32513, 33025, 33541, 34061, 35645, 36181, 37813, 38365, 40045, 40613, 41761, 42341, 42925, 44105, 44701, 45301, 47125, 47741, 49613, 50245, 52165, 52813, 54121, 54781, 55445, 56785, 57461, 58141, 60205, 60901, 63013, 63725, 65161, 65885, 66613, 68821, 69565, 71821, 72581, 74885, 75661, 78013, 78805, 81205, 82013, 84461, 85285, 87781, 88621, 91165, 92021, 94613, 95485, 97241, 98125, 99013, 99905, 101701, 102605, 105341, 106261, 109045, 109981, 112813, 113765, 114721, 115681, 116645, 117613, 120541, 121525, 122513, 124501, 125501, 128525, 129541, 130561, 132613, 133645, 136765, 137813, 139921, 140981, 142045, 144185, 145261, 146341, 149605, 150701, 154013, 155125, 158485, 159613, 163021, 164165, 166465, 167621, 168781, 169945, 171113, 172285, 173461, 174641, 177013, 178205, 181805, 183013, 186661, 187885, 191581, 192821, 195313, 196565, 197821, 201613, 202885, 206725, 208013, 211901, 213205, 217141, 218461, 219785, 222445, 223781, 227813, 229165, 230521, 233245, 234613, 237361, 238741, 240125, 244301, 245701, 249925, 251341, 255613, 257045, 261365, 262813, 265721, 267181, 268645, 270113, 273061, 274541, 279005, 280501, 285013, 286525, 291085, 292613, 297221, 298765, 303421, 304981, 306545, 309685, 311261, 316013, 317605, 319201, 322405, 324013, 328861, 330485, 332113, 333745, 335381, 337021, 341965, 343621, 348613, 350285, 353641, 355325, 357013, 362101, 363805, 368941, 370661, 375845, 377581, 379321, 382813, 384565, 389845, 391613, 395161, 396941, 398725, 404101, 405901.

Some larger examples :