等分正方形边长形成的交点数和区域数

wayne

mathe 发表于 2025-4-16 21:58
下面这段C++代码，通过修改宏N,就可以计算第N项的结果。

根据mathe的代码，用80GB内存的电脑继续接力，计算到了198,199的时候,80GB内存扛不下去了.

1. {160, 5041691581, 5202490088, {2 -> 4932369180, 3 -> 83633308, 4 -> 15450824, 5 -> 5058896, 6 -> 2193192, 7 -> 1133244, 8 -> 618528, 9 -> 378408, 10 -> 238016, 11 -> 151832, 12 -> 108912, 13 -> 81348, 14 -> 57184, 15 -> 43480, 16 -> 32008, 17 -> 20464, 18 -> 18440, 19 -> 16756, 20 -> 11972, 21 -> 9936, 22 -> 7440, 23 -> 7920, 24 -> 8616, 25 -> 5524, 26 -> 3988, 27 -> 2852, 28 -> 2456, 29 -> 2028, 30 -> 1720, 31 -> 1444, 32 -> 2336, 33 -> 2980, 34 -> 1820, 35 -> 1264, 36 -> 884, 37 -> 736, 38 -> 632, 39 -> 640, 40 -> 584, 41 -> 956, 42 -> 824, 43 -> 816, 44 -> 448, 45 -> 400, 46 -> 248, 47 -> 212, 48 -> 276, 49 -> 168, 50 -> 204, 51 -> 108, 52 -> 168, 53 -> 132, 54 -> 364, 55 -> 704, 56 -> 476, 57 -> 220, 58 -> 148, 59 -> 140, 60 -> 56, 61 -> 72, 62 -> 76, 63 -> 60, 64 -> 104, 65 -> 40, 66 -> 68, 67 -> 36, 68 -> 72, 69 -> 32, 70 -> 52, 71 -> 60, 72 -> 36, 73 -> 48, 74 -> 16, 75 -> 28, 76 -> 24, 77 -> 24, 78 -> 36, 79 -> 36, 80 -> 24, 81 -> 148, 82 -> 132, 83 -> 140, 84 -> 68, 85 -> 104, 86 -> 40, 87 -> 40, 88 -> 36, 89 -> 36, 91 -> 24, 92 -> 16, 93 -> 4, 94 -> 4, 95 -> 16, 97 -> 12, 98 -> 8, 99 -> 8, 100 -> 8, 103 -> 16, 104 -> 4, 105 -> 4, 106 -> 4, 108 -> 8, 109 -> 4, 111 -> 8, 112 -> 8, 114 -> 8, 115 -> 8, 116 -> 4, 118 -> 4, 119 -> 4, 120 -> 4, 121 -> 8, 122 -> 12, 123 -> 4, 124 -> 8, 125 -> 4, 126 -> 4, 128 -> 4, 129 -> 4, 130 -> 4, 131 -> 8, 132 -> 4, 133 -> 4, 135 -> 4, 138 -> 8, 139 -> 4, 140 -> 4, 141 -> 8, 142 -> 12, 143 -> 4, 145 -> 4, 146 -> 4, 148 -> 4, 149 -> 4, 150 -> 4, 151 -> 8, 153 -> 4, 154 -> 4, 155 -> 4, 158 -> 4, 159 -> 4, 160 -> 4, 161 -> 8, 162 -> 24, 163 -> 4, 184 -> 4, 320 -> 1}}
   
2. {161, 5289185077, 5411567160, {2 -> 5205949184, 3 -> 64090656, 4 -> 11368108, 5 -> 3818368, 6 -> 1662408, 7 -> 856364, 8 -> 462612, 9 -> 293140, 10 -> 189532, 11 -> 116632, 12 -> 82884, 13 -> 64596, 14 -> 45636, 15 -> 37324, 16 -> 26520, 17 -> 16908, 18 -> 17276, 19 -> 12736, 20 -> 10740, 21 -> 8800, 22 -> 5148, 23 -> 8344, 24 -> 7656, 25 -> 3744, 26 -> 2640, 27 -> 2060, 28 -> 2816, 29 -> 1860, 30 -> 1052, 31 -> 824, 32 -> 1560, 33 -> 2884, 34 -> 2052, 35 -> 1188, 36 -> 700, 37 -> 428, 38 -> 424, 39 -> 356, 40 -> 432, 41 -> 1100, 42 -> 844, 43 -> 532, 44 -> 376, 45 -> 260, 46 -> 188, 47 -> 164, 48 -> 168, 49 -> 60, 50 -> 92, 51 -> 140, 52 -> 120, 53 -> 60, 54 -> 480, 55 -> 808, 56 -> 448, 57 -> 272, 58 -> 96, 59 -> 100, 60 -> 16, 61 -> 32, 62 -> 36, 63 -> 20, 64 -> 36, 65 -> 36, 66 -> 24, 67 -> 16, 68 -> 4, 69 -> 40, 70 -> 28, 71 -> 40, 72 -> 12, 73 -> 8, 74 -> 12, 75 -> 32, 76 -> 24, 77 -> 28, 78 -> 16, 79 -> 4, 80 -> 24, 81 -> 112, 82 -> 280, 83 -> 152, 84 -> 72, 85 -> 32, 86 -> 48, 87 -> 12, 88 -> 4, 89 -> 4, 91 -> 4, 92 -> 20, 93 -> 12, 95 -> 4, 98 -> 8, 99 -> 12, 100 -> 4, 102 -> 4, 103 -> 4, 104 -> 8, 105 -> 4, 106 -> 4, 107 -> 16, 109 -> 4, 110 -> 4, 112 -> 4, 113 -> 8, 114 -> 4, 116 -> 4, 117 -> 8, 118 -> 4, 119 -> 4, 120 -> 4, 123 -> 4, 124 -> 4, 125 -> 8, 126 -> 4, 128 -> 4, 129 -> 8, 130 -> 8, 132 -> 4, 133 -> 8, 134 -> 4, 135 -> 4, 136 -> 4, 138 -> 4, 139 -> 8, 140 -> 4, 141 -> 4, 142 -> 8, 143 -> 4, 144 -> 4, 145 -> 4, 146 -> 4, 147 -> 4, 148 -> 4, 149 -> 4, 151 -> 4, 152 -> 4, 153 -> 4, 154 -> 4, 155 -> 4, 156 -> 8, 157 -> 4, 158 -> 4, 159 -> 4, 160 -> 4, 161 -> 8, 162 -> 4, 322 -> 1}}
   
3. {162, 5340749393, 5495151944, {2 -> 5236096224, 3 -> 79709112, 4 -> 15032124, 5 -> 4988452, 6 -> 2094244, 7 -> 1052452, 8 -> 584376, 9 -> 364312, 10 -> 233612, 11 -> 149748, 12 -> 103728, 13 -> 75376, 14 -> 51468, 15 -> 43100, 16 -> 30780, 17 -> 19716, 18 -> 18660, 19 -> 15368, 20 -> 11796, 21 -> 10936, 22 -> 7740, 23 -> 7648, 24 -> 7024, 25 -> 5444, 26 -> 3536, 27 -> 3100, 28 -> 2632, 29 -> 2284, 30 -> 1700, 31 -> 1476, 32 -> 1468, 33 -> 2564, 34 -> 2748, 35 -> 1476, 36 -> 1140, 37 -> 644, 38 -> 528, 39 -> 480, 40 -> 588, 41 -> 740, 42 -> 812, 43 -> 640, 44 -> 580, 45 -> 488, 46 -> 396, 47 -> 352, 48 -> 176, 49 -> 180, 50 -> 104, 51 -> 176, 52 -> 120, 53 -> 116, 54 -> 308, 55 -> 500, 56 -> 332, 57 -> 468, 58 -> 260, 59 -> 172, 60 -> 164, 61 -> 112, 62 -> 84, 63 -> 24, 64 -> 60, 65 -> 56, 66 -> 24, 67 -> 16, 68 -> 52, 69 -> 24, 70 -> 24, 71 -> 32, 72 -> 32, 73 -> 20, 74 -> 20, 75 -> 16, 76 -> 16, 77 -> 40, 78 -> 8, 79 -> 20, 80 -> 36, 81 -> 20, 82 -> 252, 83 -> 236, 85 -> 44, 86 -> 32, 87 -> 12, 88 -> 20, 89 -> 8, 90 -> 48, 91 -> 68, 92 -> 16, 93 -> 8, 94 -> 8, 96 -> 8, 97 -> 16, 98 -> 20, 99 -> 4, 100 -> 16, 101 -> 4, 102 -> 12, 103 -> 4, 104 -> 4, 105 -> 4, 106 -> 4, 107 -> 4, 109 -> 8, 110 -> 12, 111 -> 4, 112 -> 12, 113 -> 4, 114 -> 4, 115 -> 8, 116 -> 8, 121 -> 4, 122 -> 8, 123 -> 4, 124 -> 8, 125 -> 8, 126 -> 4, 127 -> 4, 129 -> 8, 132 -> 4, 133 -> 8, 134 -> 8, 137 -> 4, 138 -> 8, 139 -> 8, 141 -> 4, 142 -> 4, 144 -> 4, 145 -> 8, 146 -> 12, 147 -> 4, 148 -> 4, 150 -> 4, 151 -> 8, 153 -> 4, 154 -> 4, 157 -> 4, 158 -> 12, 159 -> 8, 162 -> 4, 163 -> 12, 166 -> 8, 170 -> 4, 180 -> 4, 192 -> 4, 324 -> 1}}
   
4. {163, 5638078485, 5737340772, {2 -> 5571372980, 3 -> 51170796, 4 -> 9041756, 5 -> 3152572, 6 -> 1384924, 7 -> 723956, 8 -> 388044, 9 -> 251056, 10 -> 161072, 11 -> 101732, 12 -> 70112, 13 -> 57700, 14 -> 33696, 15 -> 38676, 16 -> 23016, 17 -> 12276, 18 -> 16952, 19 -> 11260, 20 -> 8500, 21 -> 8324, 22 -> 4236, 23 -> 7372, 24 -> 7984, 25 -> 3332, 26 -> 932, 27 -> 2444, 28 -> 2540, 29 -> 1308, 30 -> 676, 31 -> 604, 32 -> 1396, 33 -> 4012, 34 -> 2508, 35 -> 236, 36 -> 212, 37 -> 276, 38 -> 228, 39 -> 220, 40 -> 204, 41 -> 1548, 42 -> 1404, 43 -> 100, 44 -> 116, 45 -> 76, 46 -> 116, 47 -> 100, 48 -> 84, 49 -> 116, 50 -> 68, 51 -> 116, 52 -> 68, 53 -> 108, 54 -> 392, 55 -> 988, 56 -> 764, 57 -> 12, 58 -> 20, 59 -> 28, 60 -> 20, 61 -> 20, 62 -> 28, 63 -> 12, 64 -> 20, 65 -> 44, 66 -> 12, 67 -> 20, 68 -> 36, 69 -> 12, 70 -> 20, 71 -> 28, 72 -> 12, 73 -> 20, 74 -> 28, 75 -> 12, 76 -> 20, 77 -> 28, 78 -> 12, 79 -> 20, 80 -> 28, 81 -> 20, 82 -> 236, 83 -> 444, 84 -> 4, 85 -> 4, 86 -> 4, 87 -> 4, 88 -> 4, 89 -> 4, 90 -> 4, 91 -> 4, 92 -> 4, 93 -> 4, 94 -> 4, 95 -> 4, 96 -> 4, 97 -> 4, 98 -> 4, 99 -> 4, 100 -> 4, 101 -> 4, 102 -> 4, 103 -> 4, 104 -> 4, 105 -> 4, 106 -> 4, 107 -> 4, 108 -> 4, 109 -> 12, 110 -> 4, 111 -> 4, 112 -> 4, 113 -> 4, 114 -> 4, 115 -> 4, 116 -> 4, 117 -> 4, 118 -> 4, 119 -> 4, 120 -> 4, 121 -> 4, 122 -> 4, 123 -> 4, 124 -> 4, 125 -> 4, 126 -> 4, 127 -> 4, 128 -> 4, 129 -> 4, 130 -> 4, 131 -> 4, 132 -> 4, 133 -> 4, 134 -> 4, 135 -> 4, 136 -> 4, 137 -> 4, 138 -> 4, 139 -> 4, 140 -> 4, 141 -> 4, 142 -> 4, 143 -> 4, 144 -> 4, 145 -> 4, 146 -> 4, 147 -> 4, 148 -> 4, 149 -> 4, 150 -> 4, 151 -> 4, 152 -> 4, 153 -> 4, 154 -> 4, 155 -> 4, 156 -> 4, 157 -> 4, 158 -> 4, 159 -> 4, 160 -> 4, 161 -> 4, 162 -> 4, 163 -> 8, 164 -> 4, 326 -> 1}}
   
5. {164, 5693409877, 5825757984, {2 -> 5604067652, 3 -> 68118228, 4 -> 12680072, 5 -> 4186232, 6 -> 1848348, 7 -> 947828, 8 -> 521420, 9 -> 316164, 10 -> 201504, 11 -> 127148, 12 -> 85412, 13 -> 67656, 14 -> 42976, 15 -> 42128, 16 -> 28488, 17 -> 17880, 18 -> 17388, 19 -> 14160, 20 -> 10064, 21 -> 9260, 22 -> 7236, 23 -> 7072, 24 -> 8004, 25 -> 5296, 26 -> 3024, 27 -> 2560, 28 -> 2592, 29 -> 2244, 30 -> 1616, 31 -> 1196, 32 -> 816, 33 -> 2952, 34 -> 2580, 35 -> 1144, 36 -> 928, 37 -> 424, 38 -> 648, 39 -> 364, 40 -> 268, 41 -> 692, 42 -> 964, 43 -> 788, 44 -> 660, 45 -> 256, 46 -> 180, 47 -> 128, 48 -> 128, 49 -> 96, 50 -> 104, 51 -> 136, 52 -> 96, 53 -> 56, 54 -> 144, 55 -> 252, 56 -> 744, 57 -> 440, 58 -> 332, 59 -> 236, 60 -> 24, 61 -> 40, 62 -> 56, 63 -> 92, 64 -> 28, 65 -> 116, 66 -> 24, 67 -> 32, 68 -> 20, 69 -> 24, 70 -> 36, 71 -> 8, 72 -> 36, 73 -> 24, 74 -> 8, 75 -> 20, 76 -> 32, 78 -> 56, 79 -> 20, 81 -> 24, 82 -> 60, 83 -> 260, 84 -> 128, 85 -> 144, 86 -> 104, 87 -> 32, 88 -> 20, 89 -> 4, 91 -> 8, 92 -> 12, 93 -> 4, 96 -> 4, 97 -> 4, 98 -> 4, 99 -> 8, 100 -> 8, 102 -> 8, 103 -> 8, 104 -> 4, 105 -> 4, 106 -> 4, 108 -> 8, 109 -> 8, 110 -> 4, 111 -> 8, 112 -> 4, 114 -> 8, 115 -> 8, 116 -> 4, 118 -> 4, 119 -> 4, 121 -> 8, 122 -> 12, 123 -> 4, 126 -> 4, 127 -> 4, 129 -> 4, 130 -> 12, 131 -> 4, 133 -> 4, 134 -> 4, 135 -> 4, 137 -> 4, 138 -> 12, 139 -> 4, 140 -> 4, 141 -> 4, 142 -> 4, 143 -> 4, 145 -> 4, 146 -> 12, 147 -> 4, 149 -> 4, 150 -> 4, 151 -> 4, 153 -> 4, 154 -> 12, 155 -> 4, 157 -> 8, 158 -> 4, 159 -> 4, 161 -> 4, 162 -> 12, 163 -> 4, 165 -> 8, 166 -> 4, 170 -> 4, 328 -> 1}}
   
6. {165, 5733035989, 5904553884, {2 -> 5616625252, 3 -> 89036576, 4 -> 16286072, 5 -> 5518380, 6 -> 2389844, 7 -> 1215564, 8 -> 657716, 9 -> 400004, 10 -> 252992, 11 -> 159740, 12 -> 104172, 13 -> 84900, 14 -> 60936, 15 -> 49824, 16 -> 37508, 17 -> 25704, 18 -> 20528, 19 -> 16628, 20 -> 12680, 21 -> 10748, 22 -> 8784, 23 -> 6972, 24 -> 7364, 25 -> 6088, 26 -> 4012, 27 -> 3628, 28 -> 3064, 29 -> 2912, 30 -> 2456, 31 -> 1900, 32 -> 1480, 33 -> 2524, 34 -> 2604, 35 -> 1772, 36 -> 1860, 37 -> 984, 38 -> 768, 39 -> 552, 40 -> 380, 41 -> 468, 42 -> 920, 43 -> 872, 44 -> 620, 45 -> 520, 46 -> 284, 47 -> 284, 48 -> 224, 49 -> 224, 50 -> 176, 51 -> 204, 52 -> 120, 53 -> 144, 54 -> 92, 55 -> 84, 56 -> 792, 57 -> 416, 58 -> 368, 59 -> 344, 60 -> 252, 61 -> 80, 62 -> 120, 63 -> 36, 64 -> 64, 65 -> 64, 66 -> 60, 67 -> 60, 68 -> 40, 69 -> 40, 70 -> 32, 71 -> 32, 72 -> 28, 74 -> 8, 75 -> 40, 76 -> 28, 77 -> 32, 78 -> 24, 79 -> 20, 80 -> 44, 81 -> 24, 82 -> 4, 83 -> 16, 84 -> 156, 85 -> 124, 86 -> 108, 87 -> 172, 88 -> 44, 89 -> 40, 90 -> 8, 91 -> 12, 92 -> 32, 93 -> 20, 94 -> 20, 95 -> 16, 96 -> 40, 97 -> 20, 99 -> 4, 100 -> 12, 102 -> 8, 104 -> 8, 105 -> 8, 106 -> 4, 107 -> 4, 109 -> 24, 110 -> 4, 112 -> 8, 113 -> 12, 115 -> 12, 116 -> 4, 118 -> 4, 119 -> 12, 122 -> 8, 123 -> 8, 124 -> 4, 127 -> 4, 128 -> 4, 129 -> 4, 130 -> 12, 131 -> 4, 132 -> 16, 133 -> 4, 134 -> 8, 135 -> 8, 136 -> 4, 137 -> 4, 139 -> 4, 140 -> 4, 141 -> 4, 142 -> 4, 143 -> 4, 146 -> 4, 147 -> 8, 149 -> 4, 150 -> 4, 151 -> 4, 152 -> 12, 153 -> 4, 154 -> 4, 155 -> 4, 156 -> 4, 157 -> 4, 158 -> 4, 159 -> 4, 160 -> 4, 161 -> 4, 162 -> 4, 163 -> 4, 164 -> 8, 165 -> 4, 166 -> 8, 330 -> 1}}
   
7. 166 6023174653 6145681096
   
8. 167 6217664353 6325473336
   
9. 168 6069707633 6285723104
   
10. 169 6476936189 6606368600
    
11. 170 6503600445 6682148816
    
12. 171 6699841709 6867844936
    
13. 172 6900028181 7056905648
    
14. 173 7166138525 7289006532
    
15. 174 7176210481 7358572352
    
16. 175 7333838053 7524141356
    
17. 176 7457713109 7668461632
    
18. 177 7782850965 7943198560
    
19. 178 7981599837 8138363160
    
20. 179 8221372133 8359989964
    
21. 180 7996941713 8283472144
    
22. 181 8596151661 8740921344
    
23. 182 8563211069 8792562880
    
24. 183 8905354785 9084861516
    
25. 184 8997732841 9218742440
    
26. 185 9284155229 9477994920
    
27. 186 9399096645 9628589360
    
28. 187 9677261369 9885692880
    
29. 188 9885688461 10098511136
    
30. 189 9963095465 10229065280
    
31. 190 10204201373 10465718408
    
32. 191 10672177017 10848515520
    
33. 192 10551294309 10855309352
    
34. 193 11129163941 11312204652
    
35. 194 11293906585 11505970216
    
36. 195 11291427453 11592684292
    
37. 196 11564358401 11857977904
    
38. 197 12087416713 12284333196
    
39. 198 11937912301 12281096472
    

复制代码

mathe

196 11564358401 11857977904
197 12087416713 12284333196
198 11937912301 12281096472
199 12588709085 12792984756
200 12457806449 12804760976
201 12999527965 13249794924
202 13290367085 13535429568
203 13491804381 13764484900
204 13508860605 13875818424
205 14047610781 14325501824
206 14383276269 14645940960
207 14513784885 14835741596
208 14678343889 15049071416
209 15160066025 15467503100
210 14961151301 15447078048
211 15928877097 16182853144
212 16051483901 16376287480
213 16422300165 16729548848
214 16768305593 17069418824
215 17025156193 17352536440
216 16921230481 17405359112
217 17651936273 17997934512
218 18066169957 18388102768
219 18370755121 18708565264
220 18277781169 18775913288
221 18985120301 19360296800
222 19216917789 19639435840
223 19901141729 20210329468
224 19720457333 20230180088
225 20118360945 20621877512
226 20884433241 21251860288
227 21375490265 21705503568
228 21197769089 21733169856
229 22143284377 22483858100
230 22110102425 22609743624
231 22389545817 22936653656
232 22959013609 23451548744
233 23738204121 24101638316
234 23495161609 24099738912
235 24359364941 24809852796
236 24737986897 25210650848
237 25249403605 25697827700
238 25352233801 25926889464
239 26294246741 26692532260
240 25690907145 26467952216
241 27187454733 27598976072
242 27310208881 27849958896
243 27759443153 28309974940
244 28297572077 28828261672
245 28582721893 29188293260
246 29086308133 29688969528
247 29725836813 30280604844
248 30054788921 30674297712
249 30804295845 31339411776
250 30992052261 31648998304
251 32016696089 32493168220
252 31369066885 32267112416

王守恩

正方形边长 n 等分 = n × n,  交点是这样一串数——A331449。
5, 37, 257, 817, 2757, 4825, 12293, 19241, 33549, 49577, 87685, 101981, 178465, 220113, 286357, 379097, 551669, 606241, 880293, 951445, 1209049, 1507521,

矩形边长 a × n 交点是这样一串数——A331453——只有703项。能再来几项？谢谢！！！
5,
13, 37,
35, 99, 257,
75, 213, 421, 817,
159, 401, 881, 1489, 2757,
275, 657, 1305, 2143, 3555, 4825,
477, 1085, 2131, 3431, 5821, 7663, 12293,
755, 1619, 2941, 4817, 7477, 9913, 15037, 19241,
1163, 2327, 4369, 6495, 10393, 13647, 20425, 24651, 33549,

正方形边长 n 等分 = n × n,  区域是这样一串数——A255011。
4, 56, 340, 1120, 3264, 6264, 13968, 22904, 38748, 58256, 95656, 120960, 192636, 246824, 323560, 425408, 587964, 682296, 932996, 1061232, 1327524, 1634488,

矩形边长 a × n 区域是这样一串数——A331452——只有703项。能再来几项？谢谢！！！
4;
16, 56;
46, 142, 340;
104, 296, 608, 1120;
214, 544, 1124, 1916, 3264;
380, 892, 1714,  2820,  4510,  6264;
648, 1436, 2678,  4304,  6888,  9360, 13968;
1028, 2136, 3764,  6024,  9132, 12308, 17758, 22904;
1562, 3066, 5412,  8126, 12396, 16592, 23604, 29374, 38748;
2256, 4272, 7118, 10792, 16226, 20896, 29488, 36812, 47050, 58256;

wayne

已知凸边界上的四个三点不共线的点,是可以唯一确定一个凸四边形,以及对应的对角线的交点的. 即内点.
所以问题归结为在正边形的边上计算一下有多少个这样的凸四边形(先不考虑不同的凸四边形的中心却重合的情况),这个应该是有准确的表达式的.
$L_n = C_{4 n}^{4}-4 (4 n-n-1) C_{n+1}^{3}-4 C_{n+1}^{4} = \frac{1}{2} n \left(17 n^3-30 n^2+19 n-4\right)$

利用前面的计算数据,发现n越大时,内点个数为$P_n$,那么其中比例系数 $0.921563<\alpha_n= \frac{P_n}{L_n}<0.955695$
-=----=======
这个问题如果换种花样的出题目,比如把正方形换成 任意三角形,或者多边形,点的分布不均匀,那就变态了. 但内核是一样的, 分两步, 第一步是凸四边形个数的统计, 第二步是去重, 统计共心四边形的情况.

wayne

画出$n-\alpha_n$图,发现一个比较有意思的现象,就是最上层的边界曲线刚好都是$n$为素数的取值,最下层的曲线刚好是在过去的数中最多的因子的那个.
这个很好理解,因为当n为素数的时候,重合点比例越低,当n的因子越多的时候,重合的比例越高.