三角形面积公式

northwolves

王守恩 发表于 2024-7-2 20:36
找不到规律！丢了。换一道！

a(03)=1, {1,1,1},

1. a = Table[
   
2.   Length@Select[IntegerPartitions[n, {3}], Total@# > 2 #[[1]] &], {n,
   
3.    30}]

复制代码

{0,0,1,0,1,1,2,1,3,2,4,3,5,4,7,5,8,7,10,8,12,10,14,12,16,14,19,16,21,19}

northwolves

$a(k)=\text{Round}[(k + 3 mod[k, 2])^2/48]$

王守恩

northwolves 发表于 2024-7-2 21:37
$a(k)=\text{Round}[(k + 3 mod[k, 2])^2/48]$

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},

每个{ }都是三角形。

王守恩

nyy 发表于 2024-7-2 08:27
你是用穷举法吗？

请你检查找个反例出来？谢谢！

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=\(\frac{\sqrt{4a^2b^2-(a^2+b^2-c^2)^2\ }}{4}\)

1, 等边三角形 ,   三角形面积=\(\frac{\sqrt{4a^2b^2-(a^2+a^2-a^2)^2\ }}{4}=\frac{\sqrt{3a^4}}{4}\)

2, 等腰三角形 ,   三角形面积=\(\frac{\sqrt{4a^2b^2-(a^2+b^2-b^2)^2\ }}{4}=\frac{\sqrt{4a^2b^2-a^4}}{4}\)

3, 直角三角形 ,   三角形面积=\(\frac{\sqrt{4a^2b^2-(a^2+b^2-c^2)^2\ }}{4}=\frac{\sqrt{4a^2b^2}}{4}\)

4, 任意三角形 ,   三角形面积=\(\frac{\sqrt{4a^2b^2-(a^2+b^2-c^2)^2\ }}{4}\)

a,b可以互换, a,b,c也可以互换，是先确定a,b,  还是先确定c, 看每个人的喜欢。就a^2+b^2-c^2来说，我是肯定先加减后加的。

这公式不会比海伦公式差吧？

northwolves

王守恩 发表于 2024-7-3 04:46
a(03)=0,
a(04)=0,
a(05)=0,

1. Table[s=Select[IntegerPartitions[n,{3}],#[[2]]+#[[3]]>#[[1]]>#[[2]]>#[[3]]&];{n,Length@s,s},{n,50}]//MatrixForm
   

复制代码

王守恩

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},

王守恩

northwolves 发表于 2024-7-2 21:30
{0,0,1,0,1,1,2,1,3,2,4,3,5,4,7,5,8,7,10,8,12,10,14,12,16,14,19,16,21,19}

{0,0,1,0,1,1,2,1,3,2,4,3,5,4,7,5,8,7,10,8,12,10,14,12,16,14,19,16,21,19,...这样也可以。

1. Table[Round[(n - a)^2/12], {n, 1, 82}, {a, -1, 0}] // Flatten

复制代码

3,  0＜a≤b≤c≤a+b——出不来(不一样)。
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},

王守恩

northwolves 发表于 2024-7-2 21:30
{0,0,1,0,1,1,2,1,3,2,4,3,5,4,7,5,8,7,10,8,12,10,14,12,16,14,19,16,21,19}

{0,0,1,0,1,1,2,1,3,2,4,3,5,4,7,5,8,7,10,8,12,10,14,12,16,14,19,16,21,19,...这样也可以。

1. Table[Round[(n - a)^2/12], {n, 2, 83}, {a, 0, 1}] // Flatten

复制代码

3,  0＜a≤b≤c≤a+b——出不来(不一样)。
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},
......
{ 0, 1, 1, 1, 2, 2, 3, 3, 4, 4, 6, 5, 7, 7, 9, 8, 11, 10, 13, 12, 15, 14, 18, 16, 20, 19, 23, 21, 26, 24, 29, 27, 32, 30, 36, 33, 39, 37, 43, 40, 47, 44, 51, 48, 55, 52, 60, 56, 64, 61, 69, 65, 74, 70, 79, 75, 84, 80, 90, 85, 95, 91, 101, 96, 107, 102, 113, 108}

1. Table[Floor[(n - a)/2] + Floor[((n - a)^2 + 6)/12] + a, {n, 34}, {a, 0, 1}] // Flatten

复制代码

王守恩

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,

northwolves

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

复制代码

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}}