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[讨论] 三角形面积公式

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发表于 2024-6-29 12:28:54 | 显示全部楼层 |阅读模式

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三角形3边为 \(\sqrt{a},\sqrt{b},\sqrt{c}\),   三角形面积=\(\frac{\sqrt{4ab - (a + b - c)^2\ }}{4}\)

三角形3边为 \(a,b,c\),   三角形面积=\(\frac{\sqrt{4a^2b^2 - (a^2 + b^2 - c^2)^2\ \ }}{4}\)
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2024-7-6 13:18:25 | 显示全部楼层
王守恩 发表于 2024-7-6 11:36
n个苹果,分成4堆。OEIS没有这些数。我也搞不出来(手工太难了)。

1,  0≤a≤b≤c≤d≤a+b,

以7为例:
  1. Table[s=Select[IntegerPartitions[n,{4}],#[[3]]+#[[4]]>=#[[1]]>#[[2]]>#[[3]]>#[[4]]&];{n,Length@s,s},{n,50}]//MatrixForm
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1        0        {}
2        0        {}
3        0        {}
4        0        {}
5        0        {}
6        0        {}
7        0        {}
8        0        {}
9        0        {}
10        0        {}
11        0        {}
12        0        {}
13        0        {}
14        1        {{5,4,3,2}}
15        0        {}
16        0        {}
17        1        {{6,5,4,2}}
18        1        {{6,5,4,3}}
19        1        {{7,5,4,3}}
20        2        {{7,6,5,2},{7,6,4,3}}
21        1        {{7,6,5,3}}
22        2        {{8,6,5,3},{7,6,5,4}}
23        3        {{8,7,6,2},{8,7,5,3},{8,6,5,4}}
24        3        {{9,6,5,4},{8,7,6,3},{8,7,5,4}}
25        3        {{9,7,6,3},{9,7,5,4},{8,7,6,4}}
26        5        {{9,8,7,2},{9,8,6,3},{9,8,5,4},{9,7,6,4},{8,7,6,5}}
27        4        {{10,7,6,4},{9,8,7,3},{9,8,6,4},{9,7,6,5}}
28        5        {{10,8,7,3},{10,8,6,4},{10,7,6,5},{9,8,7,4},{9,8,6,5}}
29        7        {{11,7,6,5},{10,9,8,2},{10,9,7,3},{10,9,6,4},{10,8,7,4},{10,8,6,5},{9,8,7,5}}
30        7        {{11,8,7,4},{11,8,6,5},{10,9,8,3},{10,9,7,4},{10,9,6,5},{10,8,7,5},{9,8,7,6}}
31        7        {{11,9,8,3},{11,9,7,4},{11,9,6,5},{11,8,7,5},{10,9,8,4},{10,9,7,5},{10,8,7,6}}
32        10        {{12,8,7,5},{11,10,9,2},{11,10,8,3},{11,10,7,4},{11,10,6,5},{11,9,8,4},{11,9,7,5},{11,8,7,6},{10,9,8,5},{10,9,7,6}}
33        9        {{12,9,8,4},{12,9,7,5},{12,8,7,6},{11,10,9,3},{11,10,8,4},{11,10,7,5},{11,9,8,5},{11,9,7,6},{10,9,8,6}}
34        11        {{13,8,7,6},{12,10,9,3},{12,10,8,4},{12,10,7,5},{12,9,8,5},{12,9,7,6},{11,10,9,4},{11,10,8,5},{11,10,7,6},{11,9,8,6},{10,9,8,7}}
35        13        {{13,9,8,5},{13,9,7,6},{12,11,10,2},{12,11,9,3},{12,11,8,4},{12,11,7,5},{12,10,9,4},{12,10,8,5},{12,10,7,6},{12,9,8,6},{11,10,9,5},{11,10,8,6},{11,9,8,7}}
36        13        {{13,10,9,4},{13,10,8,5},{13,10,7,6},{13,9,8,6},{12,11,10,3},{12,11,9,4},{12,11,8,5},{12,11,7,6},{12,10,9,5},{12,10,8,6},{12,9,8,7},{11,10,9,6},{11,10,8,7}}
37        14        {{14,9,8,6},{13,11,10,3},{13,11,9,4},{13,11,8,5},{13,11,7,6},{13,10,9,5},{13,10,8,6},{13,9,8,7},{12,11,10,4},{12,11,9,5},{12,11,8,6},{12,10,9,6},{12,10,8,7},{11,10,9,7}}
38        18        {{14,10,9,5},{14,10,8,6},{14,9,8,7},{13,12,11,2},{13,12,10,3},{13,12,9,4},{13,12,8,5},{13,12,7,6},{13,11,10,4},{13,11,9,5},{13,11,8,6},{13,10,9,6},{13,10,8,7},{12,11,10,5},{12,11,9,6},{12,11,8,7},{12,10,9,7},{11,10,9,8}}
39        17        {{15,9,8,7},{14,11,10,4},{14,11,9,5},{14,11,8,6},{14,10,9,6},{14,10,8,7},{13,12,11,3},{13,12,10,4},{13,12,9,5},{13,12,8,6},{13,11,10,5},{13,11,9,6},{13,11,8,7},{13,10,9,7},{12,11,10,6},{12,11,9,7},{12,10,9,8}}
40        19        {{15,10,9,6},{15,10,8,7},{14,12,11,3},{14,12,10,4},{14,12,9,5},{14,12,8,6},{14,11,10,5},{14,11,9,6},{14,11,8,7},{14,10,9,7},{13,12,11,4},{13,12,10,5},{13,12,9,6},{13,12,8,7},{13,11,10,6},{13,11,9,7},{13,10,9,8},{12,11,10,7},{12,11,9,8}}
41        22        {{15,11,10,5},{15,11,9,6},{15,11,8,7},{15,10,9,7},{14,13,12,2},{14,13,11,3},{14,13,10,4},{14,13,9,5},{14,13,8,6},{14,12,11,4},{14,12,10,5},{14,12,9,6},{14,12,8,7},{14,11,10,6},{14,11,9,7},{14,10,9,8},{13,12,11,5},{13,12,10,6},{13,12,9,7},{13,11,10,7},{13,11,9,8},{12,11,10,8}}
42        23        {{16,10,9,7},{15,12,11,4},{15,12,10,5},{15,12,9,6},{15,12,8,7},{15,11,10,6},{15,11,9,7},{15,10,9,8},{14,13,12,3},{14,13,11,4},{14,13,10,5},{14,13,9,6},{14,13,8,7},{14,12,11,5},{14,12,10,6},{14,12,9,7},{14,11,10,7},{14,11,9,8},{13,12,11,6},{13,12,10,7},{13,12,9,8},{13,11,10,8},{12,11,10,9}}
43        24        {{16,11,10,6},{16,11,9,7},{16,10,9,8},{15,13,12,3},{15,13,11,4},{15,13,10,5},{15,13,9,6},{15,13,8,7},{15,12,11,5},{15,12,10,6},{15,12,9,7},{15,11,10,7},{15,11,9,8},{14,13,12,4},{14,13,11,5},{14,13,10,6},{14,13,9,7},{14,12,11,6},{14,12,10,7},{14,12,9,8},{14,11,10,8},{13,12,11,7},{13,12,10,8},{13,11,10,9}}
44        29        {{17,10,9,8},{16,12,11,5},{16,12,10,6},{16,12,9,7},{16,11,10,7},{16,11,9,8},{15,14,13,2},{15,14,12,3},{15,14,11,4},{15,14,10,5},{15,14,9,6},{15,14,8,7},{15,13,12,4},{15,13,11,5},{15,13,10,6},{15,13,9,7},{15,12,11,6},{15,12,10,7},{15,12,9,8},{15,11,10,8},{14,13,12,5},{14,13,11,6},{14,13,10,7},{14,13,9,8},{14,12,11,7},{14,12,10,8},{14,11,10,9},{13,12,11,8},{13,12,10,9}}
45        28        {{17,11,10,7},{17,11,9,8},{16,13,12,4},{16,13,11,5},{16,13,10,6},{16,13,9,7},{16,12,11,6},{16,12,10,7},{16,12,9,8},{16,11,10,8},{15,14,13,3},{15,14,12,4},{15,14,11,5},{15,14,10,6},{15,14,9,7},{15,13,12,5},{15,13,11,6},{15,13,10,7},{15,13,9,8},{15,12,11,7},{15,12,10,8},{15,11,10,9},{14,13,12,6},{14,13,11,7},{14,13,10,8},{14,12,11,8},{14,12,10,9},{13,12,11,9}}
46        31        {{17,12,11,6},{17,12,10,7},{17,12,9,8},{17,11,10,8},{16,14,13,3},{16,14,12,4},{16,14,11,5},{16,14,10,6},{16,14,9,7},{16,13,12,5},{16,13,11,6},{16,13,10,7},{16,13,9,8},{16,12,11,7},{16,12,10,8},{16,11,10,9},{15,14,13,4},{15,14,12,5},{15,14,11,6},{15,14,10,7},{15,14,9,8},{15,13,12,6},{15,13,11,7},{15,13,10,8},{15,12,11,8},{15,12,10,9},{14,13,12,7},{14,13,11,8},{14,13,10,9},{14,12,11,9},{13,12,11,10}}
47        35        {{18,11,10,8},{17,13,12,5},{17,13,11,6},{17,13,10,7},{17,13,9,8},{17,12,11,7},{17,12,10,8},{17,11,10,9},{16,15,14,2},{16,15,13,3},{16,15,12,4},{16,15,11,5},{16,15,10,6},{16,15,9,7},{16,14,13,4},{16,14,12,5},{16,14,11,6},{16,14,10,7},{16,14,9,8},{16,13,12,6},{16,13,11,7},{16,13,10,8},{16,12,11,8},{16,12,10,9},{15,14,13,5},{15,14,12,6},{15,14,11,7},{15,14,10,8},{15,13,12,7},{15,13,11,8},{15,13,10,9},{15,12,11,9},{14,13,12,8},{14,13,11,9},{14,12,11,10}}
48        36        {{18,12,11,7},{18,12,10,8},{18,11,10,9},{17,14,13,4},{17,14,12,5},{17,14,11,6},{17,14,10,7},{17,14,9,8},{17,13,12,6},{17,13,11,7},{17,13,10,8},{17,12,11,8},{17,12,10,9},{16,15,14,3},{16,15,13,4},{16,15,12,5},{16,15,11,6},{16,15,10,7},{16,15,9,8},{16,14,13,5},{16,14,12,6},{16,14,11,7},{16,14,10,8},{16,13,12,7},{16,13,11,8},{16,13,10,9},{16,12,11,9},{15,14,13,6},{15,14,12,7},{15,14,11,8},{15,14,10,9},{15,13,12,8},{15,13,11,9},{15,12,11,10},{14,13,12,9},{14,13,11,10}}
49        38        {{19,11,10,9},{18,13,12,6},{18,13,11,7},{18,13,10,8},{18,12,11,8},{18,12,10,9},{17,15,14,3},{17,15,13,4},{17,15,12,5},{17,15,11,6},{17,15,10,7},{17,15,9,8},{17,14,13,5},{17,14,12,6},{17,14,11,7},{17,14,10,8},{17,13,12,7},{17,13,11,8},{17,13,10,9},{17,12,11,9},{16,15,14,4},{16,15,13,5},{16,15,12,6},{16,15,11,7},{16,15,10,8},{16,14,13,6},{16,14,12,7},{16,14,11,8},{16,14,10,9},{16,13,12,8},{16,13,11,9},{16,12,11,10},{15,14,13,7},{15,14,12,8},{15,14,11,9},{15,13,12,9},{15,13,11,10},{14,13,12,10}}
50        44        {{19,12,11,8},{19,12,10,9},{18,14,13,5},{18,14,12,6},{18,14,11,7},{18,14,10,8},{18,13,12,7},{18,13,11,8},{18,13,10,9},{18,12,11,9},{17,16,15,2},{17,16,14,3},{17,16,13,4},{17,16,12,5},{17,16,11,6},{17,16,10,7},{17,16,9,8},{17,15,14,4},{17,15,13,5},{17,15,12,6},{17,15,11,7},{17,15,10,8},{17,14,13,6},{17,14,12,7},{17,14,11,8},{17,14,10,9},{17,13,12,8},{17,13,11,9},{17,12,11,10},{16,15,14,5},{16,15,13,6},{16,15,12,7},{16,15,11,8},{16,15,10,9},{16,14,13,7},{16,14,12,8},{16,14,11,9},{16,13,12,9},{16,13,11,10},{15,14,13,8},{15,14,12,9},{15,14,11,10},{15,13,12,10},{14,13,12,11}}
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2024-7-4 11:08:42 | 显示全部楼层
拓展成8道题(手工计算,错误难免)。可以把题目看作n个苹果,分成3堆。

1,  0≤a≤b≤c≤a+b,
a(01)=0,
a(02)=1, {0,1,1},
a(03)=1, {1,1,1},
a(04)=2, {0,2,2},{1,1,2},
a(05)=1, {1,2,2},
a(06)=3, {0,3,3},{1,2,3},{2,2,2},
a(07)=2, {1,3,3},{2,2,3},
a(08)=4, {0,4,4},{1,3,4},{2,2,4},{2,3,3}
a(09)=3, {1,4,4},{2,3,4},{3,3,3},
a(10)=5, {0,5,5},{1,4,5},{2,3,5},{2,4,4},{3,3,4},
a(11)=4, {1,5,5},{2,4,5},{3,3,5},{3,4,4},
a(12)=7, {0,6,6},{1,5,6},{2,4,6},{2,5,5},{3,3,6},{3,4,5},{4,4,4},
a(13)=5, {1,6,6},{2,5,6},{3,4,6},{3,5,5},{4,4,5},
a(14)=8, {0,7,7},{1,6,7},{2,5,7},{2,6,6},{3,4,7},{3,5,6},{4,4,6},{4,5,5},
a(15)=7, {1,7,7},{2,6,7},{3,5,7},{3,6,6},{4,4,7},{4,5,6},{5,5,5},
a(16)=0, {0,8,8},{1,7,8},{2,6,8},{2,7,7},{3,5,8},{3,6,7},{4,4,8},{4,5,7},{4,6,6},{5,5,6},
a(17)=8, {1,8,8},{2,7,8},{3,6,8},{3,7,7},{4,5,8},{4,6,7},{5,5,7},{5,6,6},
a(18)=2, {0,9,9},{1,8,9},{2,7,9},{2,8,8},{3,6,9},{3,7,8},{4,5,9},{4,6,8},{4,7,7},{5,5,8},{5,6,7}{6,6,6},
a(19)=0, {1,9,9},{2,8,9},{3,7,9},{3,8,8},{4,6,9},{4,7,8},{5,5,9},{5,6,8},{5,7,7},{6,6,7},

2,  0≤a≤b≤c<a+b,
a(01)=0,
a(02)=0,
a(03)=1, {1,1,1},
a(04)=0,
a(05)=1, {1,2,2},
a(06)=1, {0,3,3},{1,2,3},{2,2,2},
a(07)=2, {1,3,3},{2,2,3},
a(08)=1, {2,3,3}
a(09)=3, {1,4,4},{2,3,4},{3,3,3},
a(10)=5, {0,5,5},{1,4,5},{2,3,5},{2,4,4},{3,3,4},
a(11)=4, {1,5,5},{2,4,5},{3,3,5},{3,4,4},
a(12)=7, {0,6,6},{1,5,6},{2,4,6},{2,5,5},{3,3,6},{3,4,5},{4,4,4},
a(13)=5, {1,6,6},{2,5,6},{3,4,6},{3,5,5},{4,4,5},
a(14)=8, {0,7,7},{1,6,7},{2,5,7},{2,6,6},{3,4,7},{3,5,6},{4,4,6},{4,5,5},
a(15)=7, {1,7,7},{2,6,7},{3,5,7},{3,6,6},{4,4,7},{4,5,6},{5,5,5},
a(16)=0, {0,8,8},{1,7,8},{2,6,8},{2,7,7},{3,5,8},{3,6,7},{4,4,8},{4,5,7},{4,6,6},{5,5,6},
a(17)=8, {1,8,8},{2,7,8},{3,6,8},{3,7,7},{4,5,8},{4,6,7},{5,5,7},{5,6,6},
a(18)=2, {0,9,9},{1,8,9},{2,7,9},{2,8,8},{3,6,9},{3,7,8},{4,5,9},{4,6,8},{4,7,7},{5,5,8},{5,6,7}{6,6,6},
a(19)=0, {1,9,9},{2,8,9},{3,7,9},{3,8,8},{4,6,9},{4,7,8},{5,5,9},{5,6,8},{5,7,7},{6,6,7},

3,  0<a≤b≤c≤a+b
a(01)=0,
a(02)=0,
a(03)=1, {1,1,1},
a(04)=1, {1,1,2},
a(05)=1, {1,2,2},
a(06)=2, {1,2,3},{2,2,2},
a(07)=2, {1,3,3},{2,2,3},
a(08)=3, {1,3,4},{2,2,4},{2,3,3}
a(09)=3, {1,4,4},{2,3,4},{3,3,3},
a(10)=4, {1,4,5},{2,3,5},{2,4,4},{3,3,4},
a(11)=4, {1,5,5},{2,4,5},{3,3,5},{3,4,4},
a(12)=6, {1,5,6},{2,4,6},{2,5,5},{3,3,6},{3,4,5},{4,4,4},
a(13)=5, {1,6,6},{2,5,6},{3,4,6},{3,5,5},{4,4,5},
a(14)=7, {1,6,7},{2,5,7},{2,6,6},{3,4,7},{3,5,6},{4,4,6},{4,5,5},
a(15)=7, {1,7,7},{2,6,7},{3,5,7},{3,6,6},{4,4,7},{4,5,6},{5,5,5},
a(16)=9, {1,7,8},{2,6,8},{2,7,7},{3,5,8},{3,6,7},{4,4,8},{4,5,7},{4,6,6},{5,5,6},
a(17)=8, {1,8,8},{2,7,8},{3,6,8},{3,7,7},{4,5,8},{4,6,7},{5,5,7},{5,6,6},
a(18)=1, {1,8,9},{2,7,9},{2,8,8},{3,6,9},{3,7,8},{4,5,9},{4,6,8},{4,7,7},{5,5,8},{5,6,7}{6,6,6},
a(19)=0, {1,9,9},{2,8,9},{3,7,9},{3,8,8},{4,6,9},{4,7,8},{5,5,9},{5,6,8},{5,7,7},{6,6,7},

4,  0<a≤b≤c<a+b
a(01)=0,
a(02)=0,
a(03)=1, {1,1,1},
a(04)=0,
a(05)=1, {1,2,2},
a(06)=1, {2,2,2},
a(07)=2, {1,3,3},{2,2,3},
a(08)=1, {2,3,3}
a(09)=3, {1,4,4},{2,3,4},{3,3,3},
a(10)=2, {2,4,4},{3,3,4},
a(11)=4, {1,5,5},{2,4,5},{3,3,5},{3,4,4},
a(12)=3, {2,5,5},{3,4,5},{4,4,4},
a(13)=5, {1,6,6},{2,5,6},{3,4,6},{3,5,5},{4,4,5},
a(14)=4, {2,6,6},{3,5,6},{4,4,6},{4,5,5},
a(15)=7, {1,7,7},{2,6,7},{3,5,7},{3,6,6},{4,4,7},{4,5,6},{5,5,5},
a(16)=5, {2,7,7},{3,6,7},{4,5,7},{4,6,6},{5,5,6},
a(17)=8, {1,8,8},{2,7,8},{3,6,8},{3,7,7},{4,5,8},{4,6,7},{5,5,7},{5,6,6},
a(18)=6, {2,8,8},{3,7,8},{4,6,8},{5,5,8},{5,6,7}{6,6,6},
a(19)=0, {1,9,9},{2,8,9},{3,7,9},{3,8,8},{4,6,9},{4,7,8},{5,5,9},{5,6,8},{5,7,7},{6,6,7},

5,  0≤a<b<c<≤a+b,
a(01)=0,
a(02)=0,
a(03)=0,
a(04)=0,
a(05)=0,
a(06)=1, {1,2,3}
a(07)=0,
a(08)=1, {1,3,4},
a(09)=1, {2,3,4},
a(10)=2, {1,4,5},{2,3,5},
a(11)=1, {2,4,5},
a(12)=3, {1,5,6},{2,4,6},{3,4,5},
a(13)=2, {2,5,6},{3,4,6},
a(14)=4, {1,6,7},{2,5,7},{3,4,7},{3,5,6},
a(15)=3, {2,6,7},{3,5,7},{4,5,6},
a(16)=5, {1,7,8},{2,6,8},{3,5,8},{3,6,7},{4,5,7},
a(17)=4, {2,7,8},{3,6,8},{4,5,8},{4,6,7},
a(18)=7, {1,8,9},{2,7,9},{3,6,9},{3,7,8},{4,5,9},{4,6,8},{5,6,7},
a(19)=5, {2,8,9},{3,7,9},{4,6,9},{4,7,8},{5,6,8},

6,  0≤a<b<c<a+b,
a(01)=0,
a(02)=0,
a(03)=0,
a(04)=0,
a(05)=0,
a(06)=0,
a(07)=0,
a(08)=0,
a(09)=1, {2,3,4},
a(10)=0,
a(11)=1, {2,4,5},
a(12)=1, {3,4,5},
a(13)=2, {2,5,6},{3,4,6},
a(14)=1, {3,5,6},
a(15)=3, {2,6,7},{3,5,7},{4,5,6},
a(16)=2, {3,6,7},{4,5,7},
a(17)=4, {2,7,8},{3,6,8},{4,5,8},{4,6,7},
a(18)=3, {3,7,8},{4,6,8},{5,6,7},
a(19)=5, {2,8,9},{3,7,9},{4,6,9},{4,7,8},{5,6,8},

7,  0<a<b<c≤a+b,
a(01)=0,
a(02)=0,
a(03)=0,
a(04)=0,
a(05)=0,
a(06)=1, {1,2,3}
a(07)=0,
a(08)=1, {1,3,4},
a(09)=1, {2,3,4},
a(10)=2, {1,4,5},{2,3,5},
a(11)=1, {2,4,5},
a(12)=3, {1,5,6},{2,4,6},{3,4,5},
a(13)=2, {2,5,6},{3,4,6},
a(14)=4, {1,6,7},{2,5,7},{3,4,7},{3,5,6},
a(15)=3, {2,6,7},{3,5,7},{4,5,6},
a(16)=5, {1,7,8},{2,6,8},{3,5,8},{3,6,7},{4,5,7},
a(17)=4, {2,7,8},{3,6,8},{4,5,8},{4,6,7},
a(18)=7, {1,8,9},{2,7,9},{3,6,9},{3,7,8},{4,5,9},{4,6,8},{5,6,7},
a(19)=5, {2,8,9},{3,7,9},{4,6,9},{4,7,8},{5,6,8},

8,  0<a<b<c<a+b,
a(01)=0,
a(02)=0,
a(03)=0,
a(04)=0,
a(05)=0,
a(06)=0,
a(07)=0,
a(08)=0,
a(09)=1, {2,3,4},
a(10)=0,
a(11)=1, {2,4,5},
a(12)=1, {3,4,5},
a(13)=2, {2,5,6},{3,4,6},
a(14)=1, {3,5,6},
a(15)=3, {2,6,7},{3,5,7},{4,5,6},
a(16)=2, {3,6,7},{4,5,7},
a(17)=4, {2,7,8},{3,6,8},{4,5,8},{4,6,7},
a(18)=3, {3,7,8},{4,6,8},{5,6,7},
a(19)=5, {2,8,9},{3,7,9},{4,6,9},{4,7,8},{5,6,8},

点评

1,2,4,5,6,7,8是同一个公式。只有3不一样。  发表于 2024-7-6 11:37
1,2,4,5,6,7是同一个公式。只有3不一样。  发表于 2024-7-5 06:17
4, 0<a≤b≤c<a+b 8, 0<a<b<c<a+b 相对还有点意义,其它就是凑数的  发表于 2024-7-4 11:31
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2024-7-6 11:36:29 | 显示全部楼层
n个苹果,分成4堆。OEIS没有这些数。我也搞不出来(手工太难了)。

1,  0≤a≤b≤c≤d≤a+b,
a(01)=0,
a(02)=0,
a(03)=1, {0,1,1,1},
a(04)=1, {1,1,1,1},
a(05)=1, {1,1,1,2},
a(06)=2, {0,2,2,2},{1,1,2,2},
a(07)=1, {1,2,2,2},
a(08)=2, {1,2,2,3},{2,2,2,2},
a(09)=3, {0,3,3,3},{1,2,3,3},{2,2,2,3},
a(10)=3, {1,3,3,3},{2,2,2,4},{2,2,3,3},
a(11)=3, {1,3,3,4},{2,2,3,4},{2,3,3,3},
a(12)=5, {0,4,4,4},{1,3,4,4},{2,2,4,4},{2,3,3,4},{3,3,3,3},
a(13)=4, {1,4,4,4},{2,3,3,5},{2,3,4,4},{3,3,3,4},
a(14)=5, {1,4,4,5},{2,3,4,5},{2,4,4,4},{3,3,3,5},{3,3,4,4},
a(15)=7, {0,5,5,5},{1,4,5,5},{2,3,5,5},{2,4,4,5},{3,3,3,6},{3,3,4,5},{3,4,4,4},
a(16)=7, {1,5,5,5},{2,4,4,6},{2,4,5,5},{3,3,4,6},{3,3,5,5},{3,4,4,5},{4,4,4,4}
a(17)=7, {1,5,5,6},{2,4,5,6},{2,5,5,5},{3,3,5,6},{3,4,5,5},{3,4,4,6},{4,4,4,5},
a(18)=0, {0,6,6,6},{1,5,6,6},{2,4,6,6},{2,5,5,6},{3,3,6,6},{3,4,4,7},{3,4,5,6},{3,5,5,5},{4,4,4,6},{4,4,5,5},
a(19)=9, {1,6,6,6},{2,5,5,7},{2,5,6,6},{3,4,5,7},{3,4,6,6},{3,5,5,6},{4,4,4,7},{4,4,5,6},{4,5,5,5},

2,  0≤a≤b≤c≤d<a+b,
a(01)=0,
a(02)=0,
a(03)=0,
a(04)=1, {1,1,1,1},
a(05)=0,
a(06)=0,
a(07)=1, {1,2,2,2},
a(08)=1, {2,2,2,2},
a(09)=1, {2,2,2,3},
a(10)=2, {1,3,3,3},{2,2,3,3},
a(11)=1, {2,3,3,3},
a(12)=2, {2,3,3,4},{3,3,3,3},
a(13)=3, {1,4,4,4},{2,3,4,4},{3,3,3,4},
a(14)=3, {2,4,4,4},{3,3,3,5},{3,3,4,4},
a(15)=4, {2,4,4,5},{3,3,3,6},{3,3,4,5},{3,4,4,4},
a(16)=5, {1,5,5,5},{2,4,5,5},{3,3,5,5},{3,4,4,5},{4,4,4,4}
a(17)=4, {2,5,5,5},{3,4,5,5},{3,4,4,6},{4,4,4,5},
a(18)=5, {2,5,5,6},{3,4,5,6},{3,5,5,5},{4,4,4,6},{4,4,5,5},
a(19)=7, {1,6,6,6},{2,5,6,6},{3,4,6,6},{3,5,5,6},{4,4,4,7},{4,4,5,6},{4,5,5,5},

3,  0<a≤b≤c≤d≤a+b,
a(01)=0,
a(02)=0,
a(03)=1, {0,1,1,1},
a(04)=1, {1,1,1,1},
a(05)=1, {1,1,1,2},
a(06)=1, {1,1,2,2},
a(07)=1, {1,2,2,2},
a(08)=2, {1,2,2,3},{2,2,2,2},
a(09)=2, {1,2,3,3},{2,2,2,3},
a(10)=3, {1,3,3,3},{2,2,2,4},{2,2,3,3},
a(11)=3, {1,3,3,4},{2,2,3,4},{2,3,3,3},
a(12)=4, {1,3,4,4},{2,2,4,4},{2,3,3,4},{3,3,3,3},
a(13)=4, {1,4,4,4},{2,3,3,5},{2,3,4,4},{3,3,3,4},
a(14)=5, {1,4,4,5},{2,3,4,5},{2,4,4,4},{3,3,3,5},{3,3,4,4},
a(15)=6, {1,4,5,5},{2,3,5,5},{2,4,4,5},{3,3,3,6},{3,3,4,5},{3,4,4,4},
a(16)=7, {1,5,5,5},{2,4,4,6},{2,4,5,5},{3,3,4,6},{3,3,5,5},{3,4,4,5},{4,4,4,4}
a(17)=7, {1,5,5,6},{2,4,5,6},{2,5,5,5},{3,3,5,6},{3,4,5,5},{3,4,4,6},{4,4,4,5},
a(18)=9, {1,5,6,6},{2,4,6,6},{2,5,5,6},{3,3,6,6},{3,4,4,7},{3,4,5,6},{3,5,5,5},{4,4,4,6},{4,4,5,5},
a(19)=9, {1,6,6,6},{2,5,5,7},{2,5,6,6},{3,4,5,7},{3,4,6,6},{3,5,5,6},{4,4,4,7},{4,4,5,6},{4,5,5,5},

4,  0<a≤b≤c≤d<a+b,

5,  0≤a<b<c<d≤a+b,

6,  0≤a<b<c<d<a+b,

7,  0<a<b<c<d≤a+b,

8,  0<a<b<c<d<a+b,
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2024-6-29 15:36:47 | 显示全部楼层
三角形三边长为√a,√b,√c,  面积=36, 这样的三角形有几个?
答案太多了,加几条:a,b,c是整数。  a,b,c两两互素。
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2024-7-1 16:10:30 | 显示全部楼层
三角形三边长为√a,√b,√c, 面积=36, 这样的三角形有几个?
答案太多了,加几条:a,b,c是整数。 a,b,c两两互素。
答案还是多,再加:a + b > c > b > a > c - b > 0
先从简单的数开始。
边长为√a,√b,√c, 面积=n, 这样的三角形有几个?
n=03,a(03)=01,{5,8,9},
n=04,a(04)=01,{5,13,16},
n=05,a(05)=01,{8,13,17},
n=06,a(06)=02,{5,29,32}, {9,17,20},
n=07,a(07)=02,{8,25,29}, {13,17,20},
n=08,a(08)=03,{5,52,53}, {13,20,29}, {16,17,25},
n=09,a(09)=04,{8,41,45}, {9,37,40}, {13,25,36}, {17,20,29},
n=10,a(10)=02,{13,32,37}, {17,25,32},
n=11,a(11)=03,{8,61,65}, {13,40,41}, {17,29,40},
n=12,a(12)=04,{5,116,117}, {9,65,68}, {16,37,45}, {17,36,41},
n=13,a(13)=04,{5,136,137}, {8,85,89}, {17,40,53}, {20,37,41}
n=14,a(14)=05,{13,61,68}, {17,49,52}, {20,41,49}, {25,32,49}, {29,32,37},
n=15,a(15)=05,{8,113,117}, {9,101,104}, {13,72,73}, {25,37,52}, {29,36,41},
n=16,a(16)=05,{16,65,73}, {17,64,65}, {20,53,61}, {25,41,64}, {29,37,52},
n=17,a(17)=05,{5,232,233}, {8,145,149}, {13,89,100}, {25,52,53}, {37,40,41},
n=18,a(18)=08,{9,145,148}, {13,100,109}, {17,80,81}, {25,52,73}, {29,45,68}, {32,41,65}, {32,45,53}, {36,37,61}
n=19,a(19)=05,{5,289,292}, {8,181,185}, {13,113,116}, {25,61,68}, {40,41,53},
n=20,a(20)=06,{13,125,128}, {16,101,109}, {17,97,100}, {25,68,73}, {29,61,64}, {37,52,53},
n=21,a(21)=11,{5,353,356}, {8,221,225}, {9,197,200}, {13,136,145}, {20,89,101}, {29,65,72}, {36,53,65}, {37,49,72}, {40,49,61}, {41,45,68}, {45,49,52},
n=22,a(22)=07,{5,388,389}, {13,149,160}, {17,116,121}, {20,97,113}, {32,61,85}, {32,65,73}, {37,53,80},
n=23,a(23)=06,{8,265,269}, {20,109,113}, {29,73,100}, {37,65,68}, {40,53,89}, {41,52,85},
n=24,a(24)=10,{5,461,464}, {9,257,260}, {13,180,181}, {16,145,153}, {17,137,144}, {29,80,101}, {36,65,89}, {37,64,85}, {41,65,68}, {45,61,64},
n=25,a(25)=09,{8,313,317}, {13,193,200}, {17,148,157}, {25,101,116}, {25,104,109}, {29,89,100}, {37,68,97}, {41,61,100}, {52,53,73}
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2024-7-2 08:27:34 | 显示全部楼层
王守恩 发表于 2024-7-1 16:10
三角形三边长为√a,√b,√c, 面积=36, 这样的三角形有几个?
答案太多了,加几条:a,b,c是整数。 a,b,c两两 ...

你是用穷举法吗?
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2024-7-2 09:33:35 | 显示全部楼层
nyy 发表于 2024-7-2 08:27
你是用穷举法吗?

我只会用这个蹩脚的公式。3#是我一个一个一个一个删出来的。可费劲了。
  1. Table[NSolve[{k == Sqrt[4 a b - (a + b - c)^2]/4, GCD[a, b] == GCD[b, c] == GCD[c, a] == 1,  a + b > c > b > a > c - b > 0}, {a, b, c}, Integers], {k, 1, 13}]
复制代码

{{}, {}, {{a -> 5, b -> 8, c -> 9}}, {{a -> 5, b -> 13, c -> 16}}, {{a -> 8, b -> 13, c -> 17}}, {{a -> 5, b -> 29, c -> 32}, {a -> 9, b -> 17, c -> 20}},
{{a -> 8, b -> 25, c -> 29}, {a -> 13, b -> 17, c -> 20}}, {{a -> 5, b -> 52,  c -> 53}, {a -> 13, b -> 20, c -> 29}, {a -> 16, b -> 17,  c -> 25}},
{{a -> 8, b -> 41, c -> 45}, {a -> 9, b -> 37,  c -> 40}, {a -> 13, b -> 25, c -> 36}, {a -> 17, b -> 20,  c -> 29}}, {{a -> 13, b -> 32, c -> 37}, {a -> 17, b -> 25, c -> 32}},
{{a -> 8, b -> 61, c -> 65}, {a -> 13, b -> 40, c -> 41}, {a -> 17, b -> 29, c -> 40}}, {{a -> 5, b -> 116, c -> 117}, {a -> 9, b -> 65, c -> 68}, {a -> 16, b -> 37,  c -> 45},
{a -> 17, b -> 36, c -> 41}}, {{a -> 5, b -> 136,  c -> 137}, {a -> 8, b -> 85, c -> 89}, {a -> 17, b -> 40,  c -> 53}, {a -> 20, b -> 37, c -> 41}}}
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2024-7-2 11:58:46 | 显示全部楼层
王守恩 发表于 2024-7-2 09:33
我只会用这个蹩脚的公式。3#是我一个一个一个一个删出来的。可费劲了。

{{}, {}, {{a -> 5, b -> 8, c - ...

为什么要一个一个一个一个删除呢?

  1. Table[s =
  2.    Values@NSolve[{16 k^2 == 4 a b - (a + b - c)^2,
  3.       GCD[a, b] == GCD[b, c] == GCD[c, a] == 1,
  4.       a + b > c > b > a > c - b}, {a, b, c}, Integers]; {k, Length@s,
  5.    s}, {k, 1, 8}] // MatrixForm
复制代码


1        0        {}
2        0        {}
3        1        {{5,8,9}}
4        1        {{5,13,16}}
5        1        {{8,13,17}}
6        2        {{5,29,32},{9,17,20}}
7        2        {{8,25,29},{13,17,20}}
8        3        {{5,52,53},{13,20,29},{16,17,25}}
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2024-7-2 12:20:07 | 显示全部楼层
northwolves 发表于 2024-7-2 11:58
为什么要一个一个一个一个删除呢?

a(03)=9,
a(04)=16,
a(05)=17,
a(06)=20,
a(07)=20,
a(08)=25,
a(09)=29,
a(10)=32,
a(11)=40,
a(12)=41,
......

得到这样一串数: 9, 16, 17, 20, 20, 25, 29, 32, 40, 41, 41, 37, 41, 52, 41, 61, 53, 53, 52, 80, 85, 64, 73, ... 还可以拉出来吗?谢谢!
  
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2024-7-2 13:54:35 | 显示全部楼层
  1. f[k_]:=Module[{s=Flatten@Table[Values@Solve[{16 k^2==4 a b-(a+b-c)^2,GCD[a,b]==GCD[b,c]==GCD[c,a]==1,a+b>c>b>a>c-b},{b,c},Integers],{a,1,(7k+1)/3}]},s[[-1]]]
  2. Table[f[k], {k, 3, 10}]
复制代码


{9, 16, 17, 20, 20, 25, 29, 32}
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2024-7-2 13:59:37 | 显示全部楼层
  1. f[k_]:=For[a=Floor[(7k+1)/3],a>0,a--,s=Values@Solve[{16 k^2==4 a b-(a+b-c)^2,GCD[a,b]==GCD[b,c]==GCD[c,a]==1,a+b>c>b>a>c-b},{b,c},Integers];If [Length@s>0,Return[Flatten[s][[-1]]]]]

  2. Table[f[k], {k, 3, 20}]
复制代码


{9, 16, 17, 20, 20, 25, 29, 32, 40, 41, 41, 37, 41, 52, 41, 61, 53, 53}

评分

参与人数 1威望 +9 金币 +9 贡献 +9 经验 +9 鲜花 +9 收起 理由
王守恩 + 9 + 9 + 9 + 9 + 9 很给力!

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毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2024-7-2 20:36:47 | 显示全部楼层
northwolves 发表于 2024-7-2 13:59
{9, 16, 17, 20, 20, 25, 29, 32, 40, 41, 41, 37, 41, 52, 41, 61, 53, 53}

找不到规律!丢了。换一道!

a(03)=1, {1,1,1},
a(04)=0,
a(05)=1, {1,2,2},
a(06)=1, {2,2,2},
a(07)=2, {1,3,3},{2,2,3},
a(08)=1, {2,3,3}
a(09)=3, {1,4,4},{2,3,4},{3,3,3},
a(10)=2, {2,4,4},{3,3,4},
a(11)=4, {1,5,5},{2,4,5},{3,3,5},{3,4,4},
a(12)=3, {2,5,5},{3,4,5},{4,4,4},
a(13)=5, {1,6,6},{2,5,6},{3,4,6},{3,5,5},{4,4,5},
a(14)=4, {2,6,6},{3,5,6},{4,4,6},{4,5,5},
a(15)=7, {1,7,7},{2,6,7},{3,5,7},{3,6,6},{4,4,7},{4,5,6},{5,5,5},
a(16)=5, {2,7,7},{3,6,7},{4,5,7},{4,6,6},{5,5,6},
a(17)=8, {1,8,8},{2,7,8},{3,6,8},{3,7,7},{4,5,8},{4,6,7},{5,5,7},{5,6,6},
a(18)=6, {2,8,8},{3,7,8},{4,6,8},{5,5,8},{5,6,7}{6,6,6},
a(19)=0, {1,9,9},{2,8,9},{3,7,9},{3,8,8},{4,6,9},{4,7,8},{5,5,9},{5,6,8},{5,7,7},{6,6,7},

每个{ }都是三角形。大数不大于另2数和。
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
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