有多少种选法？

northwolves · 发表于 2023-10-6 22:28:37

n Max Detail
----------------------------------------------
1 76 1,3(p):76,58,56,47,21,32
2 7 2,6(p):7,6,5,4,4,2
3 8 3,9(p):8,7,4,3,6,6
4 16 4,12(p):16,14,13,6,5,2
5 8 5,15(p):8,7,6,5,4,4
6 16 6,18(p):16,11,7,7,11,9
7 10 7,21(p):10,8,4,8,7,6
8 7 8,24(p):7,7,6,4,7,7
9 40 9,27(p):40,40,38,2,3,4
10 11 10,30(p):11,8,8,7,6,4
11 27 11,33(p):27,25,20,8,8,6
12 12 12,36(p):12,12,10,6,4,7
13 308 13,39(p):308,224,194,85,127,48
14 13 14,42(p):13,12,8,4,9,8
15 84 15,45(p):84,62,56,56,53,7
16 9 16,48(p):9,9,8,6,7,7
17 128 17,51(p):128,127,98,5,48,44
18 17 18,54(p):17,12,10,11,10,4
19 28 19,57(p):28,24,10,22,19,16
20 16 20,60(p):16,14,8,10,12,7
21 16 21,63(p):16,11,10,9,8,8
22 16 22,66(p):16,13,7,7,13,9
23 34 23,69(p):34,32,23,8,12,14
24 12 24,72(p):12,9,9,9,9,6
25 20 25,75(p):20,15,13,10,8,8
26 17 26,78(p):17,17,7,5,15,16
27 91 27,81(p):91,88,80,6,63,66
28 12 28,84(p):12,10,8,8,10,11
29
30 14 30,90(p):14,12,12,11,8,6
31 31 31,93(p):31,29,28,8,6,4
32 12 32,96(p):12,11,10,7,10,9
33 14 33,99(p):14,12,11,8,8,10
34 68 34,102(p):68,56,55,22,21,2
35 14 35,105(p):14,14,12,7,8,11
36 21 36,108(p):21,16,14,13,10,6
37 55 37,111(p):55,40,38,21,20,4
38 19 38,114(p):19,19,5,19,15,16
39 32 39,117(p):32,30,24,11,17,24
40 12 40,120(p):12,11,9,11,9,10
41
42 16 42,126(p):16,13,9,11,11,13
43
44 14 44,132(p):14,14,8,11,10,12
45 24 45,135(p):24,18,16,12,10,7
46 30 46,138(p):30,26,23,16,16,26
47 215 47,141(p):215,146,144,72,77,8
48 14 48,144(p):14,14,9,8,13,13
49 19 49,147(p):19,16,12,7,14,14
50 35 50,150(p):35,31,11,21,29,32
51 73 51,153(p):73,68,43,56,48,26
52 15 52,156(p):15,13,13,11,8,13
53
54 16 54,162(p):16,15,8,14,12,11
55 20 55,165(p):20,18,14,13,16,7
56 12 56,168(p):12,11,11,11,11,12
57 49 57,171(p):49,35,30,24,22,8
58 23 58,174(p):23,21,13,7,16,17
59 145 59,177(p):145,132,131,16,16,2
60 15 60,180(p):15,15,8,15,11,13
61
62 35 62,186(p):35,32,8,6,31,28
63 24 63,189(p):24,24,13,6,16,17
64 14 64,192(p):14,14,8,12,14,14
65 26 65,195(p):26,24,16,11,16,10
66 22 66,198(p):22,19,16,14,16,6
67
68 23 68,204(p):23,21,17,8,11,13
69 38 69,207(p):38,29,24,14,16,7
70 16 70,210(p):16,14,10,12,14,11
71 28 71,213(p):28,28,22,17,11,24
72 22 72,216(p):22,18,16,13,12,7
73
74 55 74,222(p):55,51,37,16,19,23
75 24 75,225(p):24,20,10,14,16,15
76 28 76,228(p):28,24,19,16,10,14
77 16 77,231(p):16,16,10,10,15,16
78 32 78,234(p):32,30,28,11,8,7
79 119 79,237(p):119,117,114,80,84,6
80 20 80,240(p):20,19,19,9,9,12
81 24 81,243(p):24,21,12,9,18,18
82 39 82,246(p):39,35,32,13,19,7
83 208 83,249(p):208,198,164,92,126,49
84 22 84,252(p):22,18,12,16,17,9
85 80 85,255(p):80,75,58,70,56,20
86 35 86,258(p):35,31,14,8,24,20
87 78 87,261(p):78,68,56,16,23,13
88 21 88,264(p):21,20,19,19,20,5
89 199 89,267(p):199,122,120,80,85,8
90 28 90,270(p):28,24,19,16,18,7
91 26 91,273(p):26,24,13,8,22,23
92 32 92,276(p):32,30,23,28,14,16
93 25 93,279(p):25,24,20,19,8,14
94 56 94,282(p):56,55,52,11,8,6
95 32 95,285(p):32,30,25,28,19,10
96 18 96,288(p):18,17,15,17,15,8
97 422 97,291(p):422,411,364,40,136,98
98 28 98,294(p):28,22,21,20,14,8
99 30 99,297(p):30,24,14,21,17,16
100 20 100,300(p):20,16,10,16,15,14
101 44 101,303(p):44,43,33,2,40,40
102 34 102,306(p):34,32,28,15,11,8
103 150 103,309(p):150,148,103,8,56,58
104 49 104,312(p):49,48,43,7,19,13
105 18 105,315(p):18,16,12,16,15,11
106 106 106,318(p):106,88,84,35,31,8
107 320 107,321(p):320,306,147,16,175,162
108 33 108,324(p):33,32,31,7,11,9
109
110 20 110,330(p):20,18,16,13,14,10
111 464 111,333(p):464,394,319,104,182,84
112 21 112,336(p):21,20,19,15,12,9
113
114 21 114,342(p):21,19,16,11,13,15
115 40 115,345(p):40,30,16,20,26,23
116 32 116,348(p):32,29,8,13,28,24
117 40 117,351(p):40,32,14,18,28,19
118 176 118,354(p):176,175,147,3,31,31
119 32 119,357(p):32,24,22,22,24,5
120 21 120,360(p):21,20,11,11,20,21
121 31 121,363(p):31,24,20,11,22,22
122 69 122,366(p):69,68,58,7,26,20
123 50 123,369(p):50,48,48,11,8,6
124 46 124,372(p):46,42,31,16,16,20
125 38 125,375(p):38,35,28,12,16,22
126 24 126,378(p):24,22,12,19,15,20
127
128 18 128,384(p):18,18,16,12,14,14
129 26 129,387(p):26,23,16,12,16,13
130 28 130,390(p):28,24,23,16,10,11
131 76 131,393(p):76,58,56,23,21,8
132 28 132,396(p):28,25,22,21,16,8
133 28 133,399(p):28,26,22,16,13,9
134 49 134,402(p):49,43,38,12,14,8
135 24 135,405(p):24,21,18,15,12,12
136 19 136,408(p):19,19,12,12,19,19
137
138 26 138,414(p):26,24,12,11,23,24
139
140 22 140,420(p):22,18,18,11,16,15
141 74 141,423(p):74,53,48,26,28,7
142 50 142,426(p):50,39,37,28,26,4
143 64 143,429(p):64,56,53,10,15,8
144 23 144,432(p):23,21,14,10,21,19
145 47 145,435(p):47,34,32,16,21,8
146
147 28 147,441(p):28,23,21,12,14,16
148 271 148,444(p):271,257,200,51,77,91
149
150 35 150,450(p):35,25,14,20,24,14
151
152 28 152,456(p):28,24,14,22,16,19
153 74 153,459(p):74,72,28,56,51,46
154 25 154,462(p):25,21,19,17,11,16
155 56 155,465(p):56,38,28,19,33,16
156 39 156,468(p):39,34,32,8,19,14
157
158 140 158,474(p):140,119,88,101,70,48
159 384 159,477(p):384,322,276,256,183,91
160 22 160,480(p):22,21,18,9,20,21
161 32 161,483(p):32,24,19,22,23,8
162 44 162,486(p):44,40,35,16,18,7
163 73 163,489(p):73,44,40,32,35,6
164 44 164,492(p):44,30,26,23,19,16
165 21 165,495(p):21,20,14,14,20,16
166 468 166,498(p):468,262,252,209,222,16
167
168 23 168,504(p):23,22,15,11,20,19
169 49 169,507(p):49,48,44,19,8,14
170 32 170,510(p):32,21,17,19,19,13
171 50 171,513(p):50,48,45,8,22,15
172 48 172,516(p):48,40,23,16,35,22
173
174 49 174,522(p):49,46,43,42,12,48
175 32 175,525(p):32,27,18,25,20,30
176 24 176,528(p):24,23,19,21,21,9
177 64 177,531(p):64,56,41,42,52,19
178 121 178,534(p):121,112,109,11,73,75
179
180 32 180,540(p):32,31,29,9,21,15
181
182 19 182,546(p):19,17,16,15,17,19
183
184 24 184,552(p):24,23,23,11,17,23
185 374 185,555(p):374,367,264,332,332,109
186 47 186,558(p):47,44,40,6,19,16
187 38 187,561(p):38,36,24,8,25,27
188 127 188,564(p):127,77,57,63,71,40
189 30 189,567(p):30,24,12,24,21,18
190 35 190,570(p):35,32,12,22,28,32
191 304 191,573(p):304,304,154,2,239,240
192 18 192,576(p):18,17,17,17,17,16
193
194 49 194,582(p):49,43,14,12,38,32
195 30 195,585(p):30,28,24,7,24,22
196 25 196,588(p):25,22,16,18,15,18
197
198 30 198,594(p):30,24,16,21,20,25
199 152 199,597(p):152,138,128,16,121,107
200 20 200,600(p):20,19,19,19,19,12
201 40 201,603(p):40,35,33,34,8,32
202 224 202,606(p):224,199,196,26,30,5
203 47 203,609(p):47,38,24,12,29,20
204 32 204,612(p):32,21,19,19,21,11
205 50 205,615(p):50,40,35,16,24,10
206 48 206,618(p):48,46,28,7,40,35
207 32 207,621(p):32,32,25,18,11,16
208 24 208,624(p):24,22,17,22,17,13
209 56 209,627(p):56,54,50,13,16,7
210 25 210,630(p):25,24,24,19,14,12
211
212 29 212,636(p):29,23,22,12,22,16
213 472 213,639(p):472,464,410,297,80,235
214
215 56 215,645(p):56,50,8,24,49,43
216 21 216,648(p):21,21,12,18,21,21
217 36 217,651(p):36,33,32,18,10,13
218 88 218,654(p):88,84,82,8,11,7
219 38 219,657(p):38,35,32,12,12,15
220 28 220,660(p):28,20,18,18,20,13
221 166 221,663(p):166,156,83,16,100,86
222 47 222,666(p):47,44,24,6,35,32
223 208 223,669(p):208,198,112,40,126,137
224 21 224,672(p):21,21,21,18,14,16
225 64 225,675(p):64,38,28,28,38,15
226 226 226,678(p):226,184,182,71,69,4
227
228 28 228,684(p):28,28,23,18,13,14
229 115 229,687(p):115,112,74,6,44,40
230 32 230,690(p):32,31,28,24,14,13
231 26 231,693(p):26,24,13,16,24,26
232 37 232,696(p):37,35,29,8,21,23
233 71 233,699(p):71,58,56,16,21,8
234 49 234,702(p):49,39,37,16,25,11
235 76 235,705(p):76,75,65,2,40,40
236 35 236,708(p):35,29,23,11,24,23
237 66 237,711(p):66,65,28,52,64,42
238 32 238,714(p):32,23,23,19,15,17
239
240 24 240,720(p):24,22,17,22,17,15
241 440 241,723(p):440,379,346,62,344,300
242 28 242,726(p):28,26,25,24,19,10
243 36 243,729(p):36,35,15,26,24,32
244 56 244,732(p):56,53,31,13,35,39
245 38 245,735(p):38,36,34,32,33,5
246 52 246,738(p):52,49,46,21,24,6
247 35 247,741(p):35,32,32,9,22,26
248 46 248,744(p):46,38,31,32,16,24
249
250 35 250,750(p):35,30,25,20,20,10
251
252 27 252,756(p):27,25,18,16,18,22
253 32 253,759(p):32,28,14,14,28,27
254 131 254,762(p):131,113,64,23,79,57
255 32 255,765(p):32,30,7,28,29,28
256 24 256,768(p):24,22,20,14,20,18
257
258 84 258,774(p):84,78,23,12,62,56
259 42 259,777(p):42,35,31,14,28,16
260 60 260,780(p):60,56,14,14,56,55
261 32 261,783(p):32,26,23,24,18,12
262 71 262,786(p):71,69,22,4,64,62
263
264 23 264,792(p):23,23,22,12,21,21
265 256 265,795(p):256,196,190,134,84,59
266 22 266,798(p):22,22,16,19,18,20
267
268 34 268,804(p):34,28,28,23,12,26
269
270 33 270,810(p):33,24,24,18,21,12
271 362 271,813(p):362,208,203,160,168,10
272 35 272,816(p):35,33,26,10,21,19
273 24 273,819(p):24,22,20,19,17,16
274 50 274,822(p):50,49,44,14,12,11
275 25 275,825(p):25,24,20,19,20,14
276 30 276,828(p):30,26,23,16,16,20
277
278 71 278,834(p):71,69,58,4,28,26
279 60 279,837(p):60,58,16,56,49,43
280 25 280,840(p):25,23,12,22,23,25
281
282 50 282,846(p):50,44,42,12,17,11
283
284 202 284,852(p):202,181,144,50,62,41
285 28 285,855(p):28,25,24,12,22,19
286 28 286,858(p):28,28,22,26,11,24
287 44 287,861(p):44,30,28,23,19,16
288 35 288,864(p):35,31,21,30,16,26
289
290 33 290,870(p):33,23,19,19,23,16
291 232 291,873(p):232,231,154,5,168,164
292 259 292,876(p):259,259,5,73,255,256
293
294 23 294,882(p):23,21,16,19,21,23
295 85 295,885(p):85,68,64,42,31,14
296 40 296,888(p):40,35,32,34,32,6
297 52 297,891(p):52,52,46,17,11,24
298 71 298,894(p):71,69,64,43,13,31
299 32 299,897(p):32,26,23,24,14,20
300 35 300,900(p):35,31,25,21,15,24

northwolves · 发表于 2023-10-6 22:29:30

整理了一下mathe版主提供的数据，n=1-300

northwolves · 发表于 2023-10-8 21:52:14

王守恩发表于 2023-9-12 10:48
真诚请教。
1个四面体, 体积=0, 6条边都是正整数, 记最长边=a，
a(1)=0,

zerov=Det[{{0,1,1,1,1},{1,0,#1,#2,#6},{1,#1,0,#3,#5},{1,#2,#3,0,#4},{1,#6,#5,#4,0}}]==0&;
p[n_]:=Select[Prepend[n]/@Select[Tuples[Range@n,{5}],#1>=#2>n-#1&&#1>=#4&&#1>=#5&&If[#1==n,#3<=#4,True]&&If[#1==#2,#4<=#5,True]&@@#&],zerov@@(#^2)&];
Table[{n,Length@p[n],p[n]},{n,12}]//MatrixForm

复制代码

1 0 {}
2 0 {}
3 0 {}
4 2 {{4,4,2,1,2,3},{4,4,2,3,4,2}}
5 1 {{5,4,3,5,4,3}}
6 3 {{6,5,4,1,4,4},{6,5,4,6,4,4},{6,5,5,4,3,3}}
7 3 {{7,4,4,2,3,4},{7,6,5,2,5,4},{7,6,5,7,4,5}}
8 19 {{8,5,5,3,4,4},{8,5,5,6,5,5},{8,6,4,3,2,6},{8,7,3,2,5,7},{8,7,3,3,3,5},{8,7,3,5,7,3},{8,7,3,7,5,7},{8,7,3,8,7,5},{8,7,5,2,3,7},{8,7,5,3,7,5},{8,7,5,5,5,3},{8,7,5,7,3,7},{8,7,5,8,7,3},{8,7,6,3,6,4},{8,7,6,8,4,6},{8,8,4,2,4,6},{8,8,4,4,7,6},{8,8,4,6,8,4},{8,8,8,3,5,7}}
9 12 {{9,6,6,4,4,5},{9,7,4,3,3,6},{9,7,4,7,7,4},{9,8,5,1,5,7},{9,8,5,9,7,5},{9,8,7,4,7,4},{9,8,7,6,3,6},{9,8,7,9,4,7},{9,9,3,1,3,8},{9,9,3,8,9,3},{9,9,6,4,6,5},{9,9,6,5,9,6}}
10 8 {{10,7,7,5,4,6},{10,8,3,2,2,8},{10,8,4,2,5,6},{10,8,6,5,5,5},{10,8,6,10,8,6},{10,8,7,6,2,8},{10,9,8,5,8,4},{10,9,8,10,4,8}}
11 15 {{11,7,6,7,7,6},{11,8,8,6,4,7},{11,9,5,3,8,9},{11,9,5,11,9,8},{11,9,7,1,7,8},{11,9,7,11,8,7},{11,9,8,3,5,9},{11,9,8,11,9,5},{11,10,3,4,7,10},{11,10,3,11,10,7},{11,10,7,4,3,10},{11,10,7,11,10,3},{11,10,9,6,9,4},{11,10,9,11,4,9},{11,11,10,3,7,10}}
12 26 {{12,9,6,4,8,5},{12,9,7,2,9,9},{12,9,7,12,9,9},{12,9,9,2,7,9},{12,9,9,7,4,8},{12,9,9,12,7,9},{12,10,4,4,5,7},{12,10,4,7,10,4},{12,10,6,5,3,9},{12,10,8,2,8,8},{12,10,8,3,10,8},{12,10,8,8,9,3},{12,10,8,12,8,8},{12,10,10,8,6,6},{12,11,2,6,8,10},{12,11,5,5,4,8},{12,11,5,8,11,5},{12,11,7,6,11,7},{12,11,7,7,6,6},{12,11,9,8,3,9},{12,11,10,7,10,4},{12,11,10,12,4,10},{12,12,3,8,10,8},{12,12,6,3,6,9},{12,12,6,7,8,5},{12,12,6,9,12,6}}

王守恩 · 发表于 2023-10-9 13:20:11

northwolves 发表于 2023-10-8 21:52
1 0 {}
2 0 {}
3 0 {}

谢谢 northwolves！几点想法。
1, {4,4,2,1,2,3}不能组成四面体, 有"1"的都不能组成四面体。
2, {6,5,5,4,3,3},{7,4,4,2,3,4},{7,6,5,2,5,4}好像不能组成四面体。
3,对这个算式的编排，我是这样想的，
把四面体看作一个平放在桌面上的三角形ABC, P是顶点,
记6条棱BC=a,CA=b,AB=c,PA=A,PB=B,PC=C,
a,b,c是同一个三角形, a,A, b,B, c,C 互为对棱。
{a,b,c,A,B,C}={1,2,3,4,5,6}
我们记 1 是最大的，为避免重复, 余2,3,4,5,6, 小数尽量排前面。
譬如：{4,2,3,4,2,4},{5,3,4,5,3,4},{6,4,4,6,4,5},{7,4,5,7,6,5},
对这个算式的编排，我一直不得要领。谢谢 northwolves！

northwolves · 发表于 2023-10-9 14:58:52

{4,4,2,1,2,3}，{6,5,5,4,3,3}可以组成四面体,
有"1"的不能组成非零体积的四面体。

northwolves · 发表于 2023-10-9 15:00:21

{4,4,2,1,2,3}， {6,5,5,4,3,3}

northwolves · 发表于 2023-10-9 15:00:21

本帖最后由 northwolves 于 2023-10-9 15:13 编辑

我们记 1 是最大的，为避免重复, 余2,3,4,5,6, 小数尽量排前面。
--------------------
这样排下来更不好看

zerov=Det[{{0,1,1,1,1},{1,0,#1,#2,#6},{1,#1,0,#3,#5},{1,#2,#3,0,#4},{1,#6,#5,#4,0}}]==0&;
p[n_]:=Select[Prepend[n]/@Select[Tuples[Range@n,{5}],n-#2<#1<=#2&&#2>=#4&&#2>=#5&&If[#2==n,#3<=#4,True]&&If[#1==#2,#4<=#5,True]&@@#&],zerov@@(#^2)&];
Table[{n,Length@p[n],p[n]},{n,7}]//MatrixForm

复制代码

1 0 {}
2 0 {}
3 0 {}
4 2 {{4,2,4,1,3,2},{4,2,4,2,3,1}}
5 1 {{5,3,4,5,3,4}}
6 3 {{6,4,5,1,4,4},{6,4,5,6,4,4},{6,5,5,4,3,3}}
7 3 {{7,4,4,2,3,4},{7,5,6,2,4,5},{7,5,6,7,5,4}}
8 19 {{8,3,7,2,7,5},{8,3,7,3,5,3},{8,3,7,5,3,7},{8,3,7,7,7,5},{8,3,7,8,5,7},{8,4,6,3,6,2},{8,4,8,2,6,4},{8,4,8,4,6,2},{8,4,8,4,6,7},{8,5,5,3,4,4},{8,5,5,6,5,5},{8,5,7,2,7,3},{8,5,7,3,5,7},{8,5,7,5,3,5},{8,5,7,7,7,3},{8,5,7,8,3,7},{8,6,7,3,4,6},{8,6,7,8,6,4},{8,8,8,3,5,7}}

王守恩 · 发表于 2023-10-9 15:03:21

本帖最后由王守恩于 2023-10-9 15:34 编辑

northwolves 发表于 2023-10-9 15:00

1个四面体, 体积=0, 6条边都是正整数, 记最长边=a，
a(1)=0,
a(2)=0,
a(3)=0,
a(4)=1, {a,b,c,A,B,C}={4,2,3,4,2,4},
a(5)=1, {a,b,c,A,B,C}={5,3,4,5,3,4},
a(6)=1, {a,b,c,A,B,C}={6,4,4,6,4,5},
a(7)=1, {a,b,c,A,B,C}={7,4,5,7,6,5}，
a(8)=1, {a,b,c,A,B,C}=
a(9)=1, {a,b,c,A,B,C}=
......
OEIS会有这串数吗？我只要这串数：0,0,0,1,1,1,1,....。
对这个算式的编排，我一直不得要领。谢谢 northwolves！

王守恩 · 发表于 2023-10-9 15:29:41

王守恩发表于 2023-10-9 15:03
1个四面体, 体积=0, 6条边都是正整数, 记最长边=a，
a(1)=0,
a(2)=0,

你的数列情况是P-ABC,顶点P不能在某个边上
就是这句话！！！最好是这样！
对这个算式的编排，我一直不得要领。谢谢 northwolves！

northwolves · 发表于 2023-10-9 15:34:11

王守恩发表于 2023-10-9 15:29
你的数列情况是P-ABC,顶点P不能在某个边上
就是这句话！！！最好是这样！
对这个算式的编排，我一直不 ...

zerov=Det[{{0,1,1,1,1},{1,0,#1,#2,#6},{1,#1,0,#3,#5},{1,#2,#3,0,#4},{1,#6,#5,#4,0}}]==0&;
p[n_]:=Select[Prepend[n]/@Select[Tuples[Range@n,{5}],n-#2<#1<=#2&&n<#4+#5&&#4<=#2<#3+#4&&#2>=#5&&If[#2==n,#3<=#4,True]&&If[#1==#2,#4<=#5,True]&@@#&],zerov@@(#^2)&];
Table[Length@p[n],{n,20}]

复制代码

{0, 0, 0, 0, 1, 1, 1, 11, 4, 2, 13, 11, 16, 12, 14, 29, 27, 20, 35, 25}

{0, 0, 0, 2, 1, 3, 3, 19, 12, 8, 15, 26, 26, 27, 35, 64, 46, 41, 48, 64}此行是43楼的结果

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