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centered square numbers
Figurate numbers of the form n2 + (n-1)2. more

The first 600 centered square numbers :
1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 265, 313, 365, 421, 481, 545, 613, 685, 761, 841, 925, 1013, 1105, 1201, 1301, 1405, 1513, 1625, 1741, 1861, 1985, 2113, 2245, 2381, 2521, 2665, 2813, 2965, 3121, 3281, 3445, 3613, 3785, 3961, 4141, 4325, 4513, 4705, 4901, 5101, 5305, 5513, 5725, 5941, 6161, 6385, 6613, 6845, 7081, 7321, 7565, 7813, 8065, 8321, 8581, 8845, 9113, 9385, 9661, 9941, 10225, 10513, 10805, 11101, 11401, 11705, 12013, 12325, 12641, 12961, 13285, 13613, 13945, 14281, 14621, 14965, 15313, 15665, 16021, 16381, 16745, 17113, 17485, 17861, 18241, 18625, 19013, 19405, 19801, 20201, 20605, 21013, 21425, 21841, 22261, 22685, 23113, 23545, 23981, 24421, 24865, 25313, 25765, 26221, 26681, 27145, 27613, 28085, 28561, 29041, 29525, 30013, 30505, 31001, 31501, 32005, 32513, 33025, 33541, 34061, 34585, 35113, 35645, 36181, 36721, 37265, 37813, 38365, 38921, 39481, 40045, 40613, 41185, 41761, 42341, 42925, 43513, 44105, 44701, 45301, 45905, 46513, 47125, 47741, 48361, 48985, 49613, 50245, 50881, 51521, 52165, 52813, 53465, 54121, 54781, 55445, 56113, 56785, 57461, 58141, 58825, 59513, 60205, 60901, 61601, 62305, 63013, 63725, 64441, 65161, 65885, 66613, 67345, 68081, 68821, 69565, 70313, 71065, 71821, 72581, 73345, 74113, 74885, 75661, 76441, 77225, 78013, 78805, 79601, 80401, 81205, 82013, 82825, 83641, 84461, 85285, 86113, 86945, 87781, 88621, 89465, 90313, 91165, 92021, 92881, 93745, 94613, 95485, 96361, 97241, 98125, 99013, 99905, 100801, 101701, 102605, 103513, 104425, 105341, 106261, 107185, 108113, 109045, 109981, 110921, 111865, 112813, 113765, 114721, 115681, 116645, 117613, 118585, 119561, 120541, 121525, 122513, 123505, 124501, 125501, 126505, 127513, 128525, 129541, 130561, 131585, 132613, 133645, 134681, 135721, 136765, 137813, 138865, 139921, 140981, 142045, 143113, 144185, 145261, 146341, 147425, 148513, 149605, 150701, 151801, 152905, 154013, 155125, 156241, 157361, 158485, 159613, 160745, 161881, 163021, 164165, 165313, 166465, 167621, 168781, 169945, 171113, 172285, 173461, 174641, 175825, 177013, 178205, 179401, 180601, 181805, 183013, 184225, 185441, 186661, 187885, 189113, 190345, 191581, 192821, 194065, 195313, 196565, 197821, 199081, 200345, 201613, 202885, 204161, 205441, 206725, 208013, 209305, 210601, 211901, 213205, 214513, 215825, 217141, 218461, 219785, 221113, 222445, 223781, 225121, 226465, 227813, 229165, 230521, 231881, 233245, 234613, 235985, 237361, 238741, 240125, 241513, 242905, 244301, 245701, 247105, 248513, 249925, 251341, 252761, 254185, 255613, 257045, 258481, 259921, 261365, 262813, 264265, 265721, 267181, 268645, 270113, 271585, 273061, 274541, 276025, 277513, 279005, 280501, 282001, 283505, 285013, 286525, 288041, 289561, 291085, 292613, 294145, 295681, 297221, 298765, 300313, 301865, 303421, 304981, 306545, 308113, 309685, 311261, 312841, 314425, 316013, 317605, 319201, 320801, 322405, 324013, 325625, 327241, 328861, 330485, 332113, 333745, 335381, 337021, 338665, 340313, 341965, 343621, 345281, 346945, 348613, 350285, 351961, 353641, 355325, 357013, 358705, 360401, 362101, 363805, 365513, 367225, 368941, 370661, 372385, 374113, 375845, 377581, 379321, 381065, 382813, 384565, 386321, 388081, 389845, 391613, 393385, 395161, 396941, 398725, 400513, 402305, 404101, 405901, 407705, 409513, 411325, 413141, 414961, 416785, 418613, 420445, 422281, 424121, 425965, 427813, 429665, 431521, 433381, 435245, 437113, 438985, 440861, 442741, 444625, 446513, 448405, 450301, 452201, 454105, 456013, 457925, 459841, 461761, 463685, 465613, 467545, 469481, 471421, 473365, 475313, 477265, 479221, 481181, 483145, 485113, 487085, 489061, 491041, 493025, 495013, 497005, 499001, 501001, 503005, 505013, 507025, 509041, 511061, 513085, 515113, 517145, 519181, 521221, 523265, 525313, 527365, 529421, 531481, 533545, 535613, 537685, 539761, 541841, 543925, 546013, 548105, 550201, 552301, 554405, 556513, 558625, 560741, 562861, 564985, 567113, 569245, 571381, 573521, 575665, 577813, 579965, 582121, 584281, 586445, 588613, 590785, 592961, 595141, 597325, 599513, 601705, 603901, 606101, 608305, 610513, 612725, 614941, 617161, 619385, 621613, 623845, 626081, 628321, 630565, 632813, 635065, 637321, 639581, 641845, 644113, 646385, 648661, 650941, 653225, 655513, 657805, 660101, 662401, 664705, 667013, 669325, 671641, 673961, 676285, 678613, 680945, 683281, 685621, 687965, 690313, 692665, 695021, 697381, 699745, 702113, 704485, 706861, 709241, 711625, 714013, 716405, 718801.

Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 22360680 values, from 1 to 999999975403441).

n\r 0  1 
2022360680 2 
30149071207453560 3 
402236068000 4 
589442728944272044721360 5 
60149071200007453560 6 
70638876500319438363887666388766 7 
80111803400001118034000 8 
9049690400049690407453560049690400 9 
100894427204472136089442720000 10 
1104065579406557840655780406557820327890406557800

A pictorial representation of the table above
motab
Imagine to divide the members of this family by a number n and compute the remainders. Should they be uniformly distributed, each remainder from 0 to n-1 would be obtained in about (1/n)-th of the cases. This outcome is represented by a white square. Reddish (resp. bluish) squares represent remainders which appear more (resp. less) frequently than 1/n.