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centered octagonal numbers
Figurate numbers of the form 4n(n-1) + 1. more

The first 600 centered octagonal numbers :
1, 9, 25, 49, 81, 121, 169, 225, 289, 361, 441, 529, 625, 729, 841, 961, 1089, 1225, 1369, 1521, 1681, 1849, 2025, 2209, 2401, 2601, 2809, 3025, 3249, 3481, 3721, 3969, 4225, 4489, 4761, 5041, 5329, 5625, 5929, 6241, 6561, 6889, 7225, 7569, 7921, 8281, 8649, 9025, 9409, 9801, 10201, 10609, 11025, 11449, 11881, 12321, 12769, 13225, 13689, 14161, 14641, 15129, 15625, 16129, 16641, 17161, 17689, 18225, 18769, 19321, 19881, 20449, 21025, 21609, 22201, 22801, 23409, 24025, 24649, 25281, 25921, 26569, 27225, 27889, 28561, 29241, 29929, 30625, 31329, 32041, 32761, 33489, 34225, 34969, 35721, 36481, 37249, 38025, 38809, 39601, 40401, 41209, 42025, 42849, 43681, 44521, 45369, 46225, 47089, 47961, 48841, 49729, 50625, 51529, 52441, 53361, 54289, 55225, 56169, 57121, 58081, 59049, 60025, 61009, 62001, 63001, 64009, 65025, 66049, 67081, 68121, 69169, 70225, 71289, 72361, 73441, 74529, 75625, 76729, 77841, 78961, 80089, 81225, 82369, 83521, 84681, 85849, 87025, 88209, 89401, 90601, 91809, 93025, 94249, 95481, 96721, 97969, 99225, 100489, 101761, 103041, 104329, 105625, 106929, 108241, 109561, 110889, 112225, 113569, 114921, 116281, 117649, 119025, 120409, 121801, 123201, 124609, 126025, 127449, 128881, 130321, 131769, 133225, 134689, 136161, 137641, 139129, 140625, 142129, 143641, 145161, 146689, 148225, 149769, 151321, 152881, 154449, 156025, 157609, 159201, 160801, 162409, 164025, 165649, 167281, 168921, 170569, 172225, 173889, 175561, 177241, 178929, 180625, 182329, 184041, 185761, 187489, 189225, 190969, 192721, 194481, 196249, 198025, 199809, 201601, 203401, 205209, 207025, 208849, 210681, 212521, 214369, 216225, 218089, 219961, 221841, 223729, 225625, 227529, 229441, 231361, 233289, 235225, 237169, 239121, 241081, 243049, 245025, 247009, 249001, 251001, 253009, 255025, 257049, 259081, 261121, 263169, 265225, 267289, 269361, 271441, 273529, 275625, 277729, 279841, 281961, 284089, 286225, 288369, 290521, 292681, 294849, 297025, 299209, 301401, 303601, 305809, 308025, 310249, 312481, 314721, 316969, 319225, 321489, 323761, 326041, 328329, 330625, 332929, 335241, 337561, 339889, 342225, 344569, 346921, 349281, 351649, 354025, 356409, 358801, 361201, 363609, 366025, 368449, 370881, 373321, 375769, 378225, 380689, 383161, 385641, 388129, 390625, 393129, 395641, 398161, 400689, 403225, 405769, 408321, 410881, 413449, 416025, 418609, 421201, 423801, 426409, 429025, 431649, 434281, 436921, 439569, 442225, 444889, 447561, 450241, 452929, 455625, 458329, 461041, 463761, 466489, 469225, 471969, 474721, 477481, 480249, 483025, 485809, 488601, 491401, 494209, 497025, 499849, 502681, 505521, 508369, 511225, 514089, 516961, 519841, 522729, 525625, 528529, 531441, 534361, 537289, 540225, 543169, 546121, 549081, 552049, 555025, 558009, 561001, 564001, 567009, 570025, 573049, 576081, 579121, 582169, 585225, 588289, 591361, 594441, 597529, 600625, 603729, 606841, 609961, 613089, 616225, 619369, 622521, 625681, 628849, 632025, 635209, 638401, 641601, 644809, 648025, 651249, 654481, 657721, 660969, 664225, 667489, 670761, 674041, 677329, 680625, 683929, 687241, 690561, 693889, 697225, 700569, 703921, 707281, 710649, 714025, 717409, 720801, 724201, 727609, 731025, 734449, 737881, 741321, 744769, 748225, 751689, 755161, 758641, 762129, 765625, 769129, 772641, 776161, 779689, 783225, 786769, 790321, 793881, 797449, 801025, 804609, 808201, 811801, 815409, 819025, 822649, 826281, 829921, 833569, 837225, 840889, 844561, 848241, 851929, 855625, 859329, 863041, 866761, 870489, 874225, 877969, 881721, 885481, 889249, 893025, 896809, 900601, 904401, 908209, 912025, 915849, 919681, 923521, 927369, 931225, 935089, 938961, 942841, 946729, 950625, 954529, 958441, 962361, 966289, 970225, 974169, 978121, 982081, 986049, 990025, 994009, 998001, 1002001, 1006009, 1010025, 1014049, 1018081, 1022121, 1026169, 1030225, 1034289, 1038361, 1042441, 1046529, 1050625, 1054729, 1058841, 1062961, 1067089, 1071225, 1075369, 1079521, 1083681, 1087849, 1092025, 1096209, 1100401, 1104601, 1108809, 1113025, 1117249, 1121481, 1125721, 1129969, 1134225, 1138489, 1142761, 1147041, 1151329, 1155625, 1159929, 1164241, 1168561, 1172889, 1177225, 1181569, 1185921, 1190281, 1194649, 1199025, 1203409, 1207801, 1212201, 1216609, 1221025, 1225449, 1229881, 1234321, 1238769, 1243225, 1247689, 1252161, 1256641, 1261129, 1265625, 1270129, 1274641, 1279161, 1283689, 1288225, 1292769, 1297321, 1301881, 1306449, 1311025, 1315609, 1320201, 1324801, 1329409, 1334025, 1338649, 1343281, 1347921, 1352569, 1357225, 1361889, 1366561, 1371241, 1375929, 1380625, 1385329, 1390041, 1394761, 1399489, 1404225, 1408969, 1413721, 1418481, 1423249, 1428025, 1432809, 1437601.

Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 15811388 values, from 1 to 999999898700625).

n\r 0  1 
2015811388 2 
35270463105409250 3 
401581138800 4 
531622786324555006324555 5 
60105409250527046300 6 
72258770451753945175390451754000 7 
8015811388000000 8 
9527046335136410035136420035136420 9 
100632455500031622780006324555 10 
1114373992874797028747982874798287479800028747980

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.