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 |
2 | 0 | 22360680 | 2 |
3 | 0 | 14907120 | 7453560 | 3 |
4 | 0 | 22360680 | 0 | 0 | 4 |
5 | 8944272 | 8944272 | 0 | 4472136 | 0 | 5 |
6 | 0 | 14907120 | 0 | 0 | 0 | 7453560 | 6 |
7 | 0 | 6388765 | 0 | 0 | 3194383 | 6388766 | 6388766 | 7 |
8 | 0 | 11180340 | 0 | 0 | 0 | 11180340 | 0 | 0 | 8 |
9 | 0 | 4969040 | 0 | 0 | 4969040 | 7453560 | 0 | 4969040 | 0 | 9 |
10 | 0 | 8944272 | 0 | 4472136 | 0 | 8944272 | 0 | 0 | 0 | 0 | 10 |
11 | 0 | 4065579 | 4065578 | 4065578 | 0 | 4065578 | 2032789 | 0 | 4065578 | 0 | 0 |
A pictorial representation of the table above
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.