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centered triangular numbers
Figurate numbers of the form 3n(n-1)/2 + 1. more

The first 600 centered triangular numbers :
1, 4, 10, 19, 31, 46, 64, 85, 109, 136, 166, 199, 235, 274, 316, 361, 409, 460, 514, 571, 631, 694, 760, 829, 901, 976, 1054, 1135, 1219, 1306, 1396, 1489, 1585, 1684, 1786, 1891, 1999, 2110, 2224, 2341, 2461, 2584, 2710, 2839, 2971, 3106, 3244, 3385, 3529, 3676, 3826, 3979, 4135, 4294, 4456, 4621, 4789, 4960, 5134, 5311, 5491, 5674, 5860, 6049, 6241, 6436, 6634, 6835, 7039, 7246, 7456, 7669, 7885, 8104, 8326, 8551, 8779, 9010, 9244, 9481, 9721, 9964, 10210, 10459, 10711, 10966, 11224, 11485, 11749, 12016, 12286, 12559, 12835, 13114, 13396, 13681, 13969, 14260, 14554, 14851, 15151, 15454, 15760, 16069, 16381, 16696, 17014, 17335, 17659, 17986, 18316, 18649, 18985, 19324, 19666, 20011, 20359, 20710, 21064, 21421, 21781, 22144, 22510, 22879, 23251, 23626, 24004, 24385, 24769, 25156, 25546, 25939, 26335, 26734, 27136, 27541, 27949, 28360, 28774, 29191, 29611, 30034, 30460, 30889, 31321, 31756, 32194, 32635, 33079, 33526, 33976, 34429, 34885, 35344, 35806, 36271, 36739, 37210, 37684, 38161, 38641, 39124, 39610, 40099, 40591, 41086, 41584, 42085, 42589, 43096, 43606, 44119, 44635, 45154, 45676, 46201, 46729, 47260, 47794, 48331, 48871, 49414, 49960, 50509, 51061, 51616, 52174, 52735, 53299, 53866, 54436, 55009, 55585, 56164, 56746, 57331, 57919, 58510, 59104, 59701, 60301, 60904, 61510, 62119, 62731, 63346, 63964, 64585, 65209, 65836, 66466, 67099, 67735, 68374, 69016, 69661, 70309, 70960, 71614, 72271, 72931, 73594, 74260, 74929, 75601, 76276, 76954, 77635, 78319, 79006, 79696, 80389, 81085, 81784, 82486, 83191, 83899, 84610, 85324, 86041, 86761, 87484, 88210, 88939, 89671, 90406, 91144, 91885, 92629, 93376, 94126, 94879, 95635, 96394, 97156, 97921, 98689, 99460, 100234, 101011, 101791, 102574, 103360, 104149, 104941, 105736, 106534, 107335, 108139, 108946, 109756, 110569, 111385, 112204, 113026, 113851, 114679, 115510, 116344, 117181, 118021, 118864, 119710, 120559, 121411, 122266, 123124, 123985, 124849, 125716, 126586, 127459, 128335, 129214, 130096, 130981, 131869, 132760, 133654, 134551, 135451, 136354, 137260, 138169, 139081, 139996, 140914, 141835, 142759, 143686, 144616, 145549, 146485, 147424, 148366, 149311, 150259, 151210, 152164, 153121, 154081, 155044, 156010, 156979, 157951, 158926, 159904, 160885, 161869, 162856, 163846, 164839, 165835, 166834, 167836, 168841, 169849, 170860, 171874, 172891, 173911, 174934, 175960, 176989, 178021, 179056, 180094, 181135, 182179, 183226, 184276, 185329, 186385, 187444, 188506, 189571, 190639, 191710, 192784, 193861, 194941, 196024, 197110, 198199, 199291, 200386, 201484, 202585, 203689, 204796, 205906, 207019, 208135, 209254, 210376, 211501, 212629, 213760, 214894, 216031, 217171, 218314, 219460, 220609, 221761, 222916, 224074, 225235, 226399, 227566, 228736, 229909, 231085, 232264, 233446, 234631, 235819, 237010, 238204, 239401, 240601, 241804, 243010, 244219, 245431, 246646, 247864, 249085, 250309, 251536, 252766, 253999, 255235, 256474, 257716, 258961, 260209, 261460, 262714, 263971, 265231, 266494, 267760, 269029, 270301, 271576, 272854, 274135, 275419, 276706, 277996, 279289, 280585, 281884, 283186, 284491, 285799, 287110, 288424, 289741, 291061, 292384, 293710, 295039, 296371, 297706, 299044, 300385, 301729, 303076, 304426, 305779, 307135, 308494, 309856, 311221, 312589, 313960, 315334, 316711, 318091, 319474, 320860, 322249, 323641, 325036, 326434, 327835, 329239, 330646, 332056, 333469, 334885, 336304, 337726, 339151, 340579, 342010, 343444, 344881, 346321, 347764, 349210, 350659, 352111, 353566, 355024, 356485, 357949, 359416, 360886, 362359, 363835, 365314, 366796, 368281, 369769, 371260, 372754, 374251, 375751, 377254, 378760, 380269, 381781, 383296, 384814, 386335, 387859, 389386, 390916, 392449, 393985, 395524, 397066, 398611, 400159, 401710, 403264, 404821, 406381, 407944, 409510, 411079, 412651, 414226, 415804, 417385, 418969, 420556, 422146, 423739, 425335, 426934, 428536, 430141, 431749, 433360, 434974, 436591, 438211, 439834, 441460, 443089, 444721, 446356, 447994, 449635, 451279, 452926, 454576, 456229, 457885, 459544, 461206, 462871, 464539, 466210, 467884, 469561, 471241, 472924, 474610, 476299, 477991, 479686, 481384, 483085, 484789, 486496, 488206, 489919, 491635, 493354, 495076, 496801, 498529, 500260, 501994, 503731, 505471, 507214, 508960, 510709, 512461, 514216, 515974, 517735, 519499, 521266, 523036, 524809, 526585, 528364, 530146, 531931, 533719, 535510, 537304, 539101.

Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 25819889 values, from 1 to 999999963228649).

n\r 0  1 
21290994412909945 2 
30258198890 3 
46454972645497364549726454972 4 
55163978103279550010327956 5 
601290994500129099440 6 
70737711107377111737711136885560 7 
832274863227487322748632274863227486322748632274863227486 8 
90172132590086066300000 9 
102581989516397800516397825819895163977005163978 10 
1104694525234726304694525000469452546945264694525

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.