把1个正方形分成4个面积都是整数的三角形

northwolves

王守恩 发表于 2023-1-6 11:04
把1个矩形(边长是整数)分成(分点在整数位置上)4个面积都是整数(>1)的三角形，周边3个面积(不同)最大者不超 ...

暂未找到比较巧妙的计算方法：
前200项：
[0, 0, 0, 1, 3, 7, 10, 16, 23, 36, 41, 60, 66, 85, 108, 132, 140, 176, 185, 227, 264, 298, 309, 372, 402, 442, 485, 553, 567, 662, 677, 743, 805, 859, 927, 1050, 1068, 1128, 1202, 1331, 1351, 1494, 1515, 1625, 1760, 1835, 1858, 2034, 2097, 2230, 2330, 2466, 2492, 2669, 2786, 2982, 3096, 3191, 3220, 3512, 3542, 3643, 3834, 4008, 4151, 4393, 4426, 4604, 4742, 5014, 5049, 5392, 5428, 5549, 5784, 5988, 6158, 6448, 6487, 6827, 7024, 7160, 7201, 7630, 7822, 7964, 8139, 8458, 8502, 8975, 9188, 9435, 9624, 9780, 9989, 10450, 10498, 10777, 11101, 11514, 11564, 11949, 12000, 12380, 12860, 13037, 13090, 13626, 13680, 14129, 14354, 14857, 14913, 15345, 15606, 15920, 16309, 16506, 16776, 17577, 17751, 17954, 18204, 18540, 18800, 19501, 19564, 19989, 20253, 20790, 20855, 21568, 21880, 22103, 22692, 23209, 23277, 23804, 23873, 24680, 24970, 25208, 25545, 26419, 26754, 26998, 27496, 27901, 27975, 28812, 28887, 29457, 29973, 30635, 30989, 31853, 31931, 32195, 32520, 33370, 33749, 34481, 34562, 35009, 35792, 36071, 36154, 37312, 37556, 38275, 38861, 39334, 39420, 40089, 40734, 41565, 41930, 42229, 42318, 43690, 43780, 44570, 44946, 45646, 46072, 46793, 47244, 47758, 48618, 49427, 49522, 50634, 50730, 51055, 51989, 52848, 52946, 54095, 54194, 55312]

northwolves

王守恩 发表于 2023-1-6 11:04
把1个矩形(边长是整数)分成(分点在整数位置上)4个面积都是整数(>1)的三角形，周边3个面积(不同)最大者不超 ...

a(18)=176,not 166:

(176, [[2, 4, 3], [2, 5, 4], [2, 6, 5], [2, 7, 6], [2, 8, 7], [2, 9, 8], [2, 10, 9], [2, 11, 10], [2, 12, 11], [2, 13, 12], [2, 14, 13], [2, 15, 14], [2, 16, 15], [2, 17, 16], [2, 18, 17], [3, 4, 6], [3, 5, 8], [3, 6, 2], [3, 6, 10], [3, 7, 12], [3, 8, 14], [3, 9, 16], [3, 10, 4], [3, 10, 18], [3, 12, 5], [3, 14, 6], [3, 16, 7], [3, 18, 8], [4, 5, 3], [4, 5, 12], [4, 6, 15], [4, 7, 5], [4, 7, 18], [4, 8, 6], [4, 9, 2], [4, 9, 7], [4, 10, 8], [4, 11, 9], [4, 12, 3], [4, 12, 10], [4, 13, 11], [4, 14, 12], [4, 15, 13], [4, 16, 14], [4, 17, 15], [4, 18, 5], [4, 18, 16], [5, 6, 2], [5, 8, 9], [5, 9, 4], [5, 10, 12], [5, 12, 2], [5, 12, 6], [5, 12, 15], [5, 14, 18], [5, 15, 8], [5, 16, 3], [5, 18, 10], [6, 7, 4], [6, 7, 10], [6, 8, 2], [6, 8, 5], [6, 8, 12], [6, 9, 14], [6, 10, 3], [6, 10, 7], [6, 10, 16], [6, 11, 8], [6, 11, 18], [6, 12, 4], [6, 12, 9], [6, 13, 10], [6, 14, 5], [6, 14, 11], [6, 15, 2], [6, 15, 12], [6, 16, 13], [6, 17, 14], [6, 18, 7], [6, 18, 15], [7, 8, 3], [7, 8, 15], [7, 9, 8], [7, 10, 2], [7, 12, 6], [7, 15, 4], [7, 15, 16], [7, 16, 9], [7, 18, 2], [8, 9, 5], [8, 9, 10], [8, 10, 3], [8, 10, 6], [8, 11, 7], [8, 12, 2], [8, 12, 15], [8, 13, 9], [8, 14, 10], [8, 15, 3], [8, 15, 6], [8, 15, 11], [8, 16, 12], [8, 17, 13], [8, 18, 4], [8, 18, 14], [9, 10, 2], [9, 10, 4], [9, 10, 14], [9, 11, 16], [9, 12, 3], [9, 12, 10], [9, 12, 18], [9, 14, 2], [9, 14, 4], [9, 15, 8], [9, 16, 5], [9, 16, 15], [9, 18, 6], [10, 11, 6], [10, 12, 4], [10, 12, 7], [10, 13, 8], [10, 14, 3], [10, 14, 9], [10, 14, 15], [10, 15, 6], [10, 16, 2], [10, 16, 11], [10, 16, 18], [10, 17, 12], [10, 18, 8], [10, 18, 13], [11, 12, 5], [11, 12, 14], [11, 14, 4], [11, 15, 12], [11, 16, 3], [11, 18, 2], [11, 18, 10], [12, 13, 7], [12, 13, 18], [12, 14, 3], [12, 14, 5], [12, 14, 8], [12, 15, 2], [12, 15, 9], [12, 15, 14], [12, 16, 4], [12, 16, 10], [12, 17, 11], [12, 18, 3], [12, 18, 5], [13, 14, 6], [13, 15, 16], [13, 16, 5], [13, 18, 4], [13, 18, 14], [14, 15, 2], [14, 15, 8], [14, 15, 12], [14, 15, 18], [14, 16, 6], [14, 16, 9], [14, 17, 10], [14, 18, 5], [14, 18, 11], [14, 18, 16], [15, 16, 3], [15, 16, 7], [15, 18, 4], [15, 18, 6], [16, 17, 9], [16, 18, 2], [16, 18, 7], [16, 18, 10], [17, 18, 8]])

王守恩

可能是我们太超前了，回到简单一点的。

直角等腰(腰长是整数)三角形，在两条腰(整数位置)上作垂线，两条垂线相交于斜边，
两条垂线把直角等腰三角形分成了3块(1个矩形与2个三角形)，要求3块面积都是整数。

当腰长=n时，有a(n)种分法。OEIS有这串数，可是没有这种说法。

northwolves

如图
$\frac{x^2}{2} ,\frac{(n-x)^2}{2} $都是整数，故$n,x$均为偶数
\begin{cases}
a(n)=0,n=2k+1\\
a(n)=[n/2],n=2k\\
\end{cases}

王守恩

谢谢 northwolves！

直角(直角边是整数)三角形，在两条直角边(整数位置)上作垂线，两条垂线相交于斜边，
两条垂线把直角三角形分成了3块(1个矩形与2个三角形)，要求3块面积都是整数。

当较长直角边(相等也可以)=n时，有a(n)种分法。OEIS没有这串数了吧？

northwolves

王守恩 发表于 2023-1-10 08:28
谢谢 northwolves！

直角(直角边是整数)三角形，在两条直角边(整数位置)上作垂线，两条垂线相交于斜边， ...

$\sqrt{\frac{mn}{2}}=\sqrt{s_1}+\sqrt{s2},2<=m<=n$的整数解的组数

王守恩

northwolves 发表于 2023-1-10 11:54
$\sqrt{\frac{mn}{2}}=\sqrt{s_1}+\sqrt{s2},2

a(04)=02：(2+2)×(1+1),(2+2)×(2+2),
a(06)=05：(2+4)×(1+2),(2+4)×(2+4),(3+3)×(2+2),(4+2)×(2+1),(4+2)×(4+2),
a(08)=08：(2+6)×(1+3),(2+6)×(2+6),(4+4)×(1+1),(4+4)×(2+2),(4+4)×(3+3),(4+4)×(4+4),(6+2)×(3+1),(6+2)×(6+2),
a(09)=02：(3+6)×(2+4),(6+3)×(4+2),
a(10)=10：(2+8)×(1+4),(2+8)×(2+8),(4+6)×(2+3),(4+6)×(4+6),(5+5)×(2+2),(5+5)×(4+4),(6+4)×(3+2),(6+4)×(6+4),(8+2)×(4+1),(8+2)×(8+2),

得到这样一串数：2, 0, 5, 0, 8, 2, 10, 0, 20, 0, 15, 8, 24, 0, 30, 0, 36, 12, 24, 0, 56, 8, 30, 14,......

一个一个一个一个数出来的，还是没领会前面的方法

northwolves

$a(n)=[(x,n-x}\times{y,m-y},{x,1,n-1},{y,1,m-1},{xy%2=0,(n-x)(m-y)%2=0,mx=ny,2<=m<=n}]$

northwolves

1. def a(n):
   
2.     s=[[x,n-x,y,m-y] for m in range(2,n+1) for x in range(1,n) for y in range(1,m) if (x*y)%2==0 and (m*n)%2==0 and m*x==n*y]
   
3.     return(n,len(s),s)
   
4. for n in range(1,31):
   
5.     print(a(n))

复制代码

(1, 0, [])
(2, 0, [])
(3, 0, [])
(4, 2, [[2, 2, 1, 1], [2, 2, 2, 2]])
(5, 0, [])
(6, 5, [[2, 4, 1, 2], [4, 2, 2, 1], [3, 3, 2, 2], [2, 4, 2, 4], [4, 2, 4, 2]])
(7, 0, [])
(8, 8, [[4, 4, 1, 1], [2, 6, 1, 3], [4, 4, 2, 2], [6, 2, 3, 1], [4, 4, 3, 3], [2, 6, 2, 6], [4, 4, 4, 4], [6, 2, 6, 2]])
(9, 2, [[3, 6, 2, 4], [6, 3, 4, 2]])
(10, 10, [[5, 5, 2, 2], [2, 8, 1, 4], [4, 6, 2, 3], [6, 4, 3, 2], [8, 2, 4, 1], [5, 5, 4, 4], [2, 8, 2, 8], [4, 6, 4, 6], [6, 4, 6, 4], [8, 2, 8, 2]])
(11, 0, [])
(12, 20, [[6, 6, 1, 1], [4, 8, 1, 2], [8, 4, 2, 1], [6, 6, 2, 2], [2, 10, 1, 5], [4, 8, 2, 4], [6, 6, 3, 3], [8, 4, 4, 2], [10, 2, 5, 1], [3, 9, 2, 6], [6, 6, 4, 4], [9, 3, 6, 2], [4, 8, 3, 6], [8, 4, 6, 3], [6, 6, 5, 5], [2, 10, 2, 10], [4, 8, 4, 8], [6, 6, 6, 6], [8, 4, 8, 4], [10, 2, 10, 2]])
(13, 0, [])
(14, 15, [[7, 7, 2, 2], [2, 12, 1, 6], [4, 10, 2, 5], [6, 8, 3, 4], [8, 6, 4, 3], [10, 4, 5, 2], [12, 2, 6, 1], [7, 7, 4, 4], [7, 7, 6, 6], [2, 12, 2, 12], [4, 10, 4, 10], [6, 8, 6, 8], [8, 6, 8, 6], [10, 4, 10, 4], [12, 2, 12, 2]])
(15, 8, [[5, 10, 2, 4], [10, 5, 4, 2], [3, 12, 2, 8], [6, 9, 4, 6], [9, 6, 6, 4], [12, 3, 8, 2], [5, 10, 4, 8], [10, 5, 8, 4]])
(16, 24, [[8, 8, 1, 1], [4, 12, 1, 3], [8, 8, 2, 2], [12, 4, 3, 1], [8, 8, 3, 3], [2, 14, 1, 7], [4, 12, 2, 6], [6, 10, 3, 5], [8, 8, 4, 4], [10, 6, 5, 3], [12, 4, 6, 2], [14, 2, 7, 1], [8, 8, 5, 5], [4, 12, 3, 9], [8, 8, 6, 6], [12, 4, 9, 3], [8, 8, 7, 7], [2, 14, 2, 14], [4, 12, 4, 12], [6, 10, 6, 10], [8, 8, 8, 8], [10, 6, 10, 6], [12, 4, 12, 4], [14, 2, 14, 2]])
(17, 0, [])
(18, 30, [[6, 12, 1, 2], [12, 6, 2, 1], [9, 9, 2, 2], [6, 12, 2, 4], [12, 6, 4, 2], [9, 9, 4, 4], [2, 16, 1, 8], [4, 14, 2, 7], [6, 12, 3, 6], [8, 10, 4, 5], [10, 8, 5, 4], [12, 6, 6, 3], [14, 4, 7, 2], [16, 2, 8, 1], [3, 15, 2, 10], [6, 12, 4, 8], [9, 9, 6, 6], [12, 6, 8, 4], [15, 3, 10, 2], [6, 12, 5, 10], [12, 6, 10, 5], [9, 9, 8, 8], [2, 16, 2, 16], [4, 14, 4, 14], [6, 12, 6, 12], [8, 10, 8, 10], [10, 8, 10, 8], [12, 6, 12, 6], [14, 4, 14, 4], [16, 2, 16, 2]])
(19, 0, [])
(20, 38, [[10, 10, 1, 1], [10, 10, 2, 2], [4, 16, 1, 4], [8, 12, 2, 3], [12, 8, 3, 2], [16, 4, 4, 1], [10, 10, 3, 3], [5, 15, 2, 6], [10, 10, 4, 4], [15, 5, 6, 2], [2, 18, 1, 9], [4, 16, 2, 8], [6, 14, 3, 7], [8, 12, 4, 6], [10, 10, 5, 5], [12, 8, 6, 4], [14, 6, 7, 3], [16, 4, 8, 2], [18, 2, 9, 1], [10, 10, 6, 6], [10, 10, 7, 7], [4, 16, 3, 12], [8, 12, 6, 9], [12, 8, 9, 6], [16, 4, 12, 3], [5, 15, 4, 12], [10, 10, 8, 8], [15, 5, 12, 4], [10, 10, 9, 9], [2, 18, 2, 18], [4, 16, 4, 16], [6, 14, 6, 14], [8, 12, 8, 12], [10, 10, 10, 10], [12, 8, 12, 8], [14, 6, 14, 6], [16, 4, 16, 4], [18, 2, 18, 2]])
(21, 12, [[7, 14, 2, 4], [14, 7, 4, 2], [7, 14, 4, 8], [14, 7, 8, 4], [3, 18, 2, 12], [6, 15, 4, 10], [9, 12, 6, 8], [12, 9, 8, 6], [15, 6, 10, 4], [18, 3, 12, 2], [7, 14, 6, 12], [14, 7, 12, 6]])
(22, 25, [[11, 11, 2, 2], [11, 11, 4, 4], [2, 20, 1, 10], [4, 18, 2, 9], [6, 16, 3, 8], [8, 14, 4, 7], [10, 12, 5, 6], [12, 10, 6, 5], [14, 8, 7, 4], [16, 6, 8, 3], [18, 4, 9, 2], [20, 2, 10, 1], [11, 11, 6, 6], [11, 11, 8, 8], [11, 11, 10, 10], [2, 20, 2, 20], [4, 18, 4, 18], [6, 16, 6, 16], [8, 14, 8, 14], [10, 12, 10, 12], [12, 10, 12, 10], [14, 8, 14, 8], [16, 6, 16, 6], [18, 4, 18, 4], [20, 2, 20, 2]])
(23, 0, [])
(24, 60, [[12, 12, 1, 1], [8, 16, 1, 2], [16, 8, 2, 1], [6, 18, 1, 3], [12, 12, 2, 2], [18, 6, 3, 1], [4, 20, 1, 5], [8, 16, 2, 4], [12, 12, 3, 3], [16, 8, 4, 2], [20, 4, 5, 1], [6, 18, 2, 6], [12, 12, 4, 4], [18, 6, 6, 2], [8, 16, 3, 6], [16, 8, 6, 3], [12, 12, 5, 5], [2, 22, 1, 11], [4, 20, 2, 10], [6, 18, 3, 9], [8, 16, 4, 8], [10, 14, 5, 7], [12, 12, 6, 6], [14, 10, 7, 5], [16, 8, 8, 4], [18, 6, 9, 3], [20, 4, 10, 2], [22, 2, 11, 1], [12, 12, 7, 7], [8, 16, 5, 10], [16, 8, 10, 5], [3, 21, 2, 14], [6, 18, 4, 12], [9, 15, 6, 10], [12, 12, 8, 8], [15, 9, 10, 6], [18, 6, 12, 4], [21, 3, 14, 2], [4, 20, 3, 15], [8, 16, 6, 12], [12, 12, 9, 9], [16, 8, 12, 6], [20, 4, 15, 3], [6, 18, 5, 15], [12, 12, 10, 10], [18, 6, 15, 5], [8, 16, 7, 14], [16, 8, 14, 7], [12, 12, 11, 11], [2, 22, 2, 22], [4, 20, 4, 20], [6, 18, 6, 18], [8, 16, 8, 16], [10, 14, 10, 14], [12, 12, 12, 12], [14, 10, 14, 10], [16, 8, 16, 8], [18, 6, 18, 6], [20, 4, 20, 4], [22, 2, 22, 2]])
(25, 8, [[5, 20, 2, 8], [10, 15, 4, 6], [15, 10, 6, 4], [20, 5, 8, 2], [5, 20, 4, 16], [10, 15, 8, 12], [15, 10, 12, 8], [20, 5, 16, 4]])
(26, 30, [[13, 13, 2, 2], [13, 13, 4, 4], [13, 13, 6, 6], [2, 24, 1, 12], [4, 22, 2, 11], [6, 20, 3, 10], [8, 18, 4, 9], [10, 16, 5, 8], [12, 14, 6, 7], [14, 12, 7, 6], [16, 10, 8, 5], [18, 8, 9, 4], [20, 6, 10, 3], [22, 4, 11, 2], [24, 2, 12, 1], [13, 13, 8, 8], [13, 13, 10, 10], [13, 13, 12, 12], [2, 24, 2, 24], [4, 22, 4, 22], [6, 20, 6, 20], [8, 18, 8, 18], [10, 16, 10, 16], [12, 14, 12, 14], [14, 12, 14, 12], [16, 10, 16, 10], [18, 8, 18, 8], [20, 6, 20, 6], [22, 4, 22, 4], [24, 2, 24, 2]])
(27, 14, [[9, 18, 2, 4], [18, 9, 4, 2], [9, 18, 4, 8], [18, 9, 8, 4], [3, 24, 2, 16], [6, 21, 4, 14], [9, 18, 6, 12], [12, 15, 8, 10], [15, 12, 10, 8], [18, 9, 12, 6], [21, 6, 14, 4], [24, 3, 16, 2], [9, 18, 8, 16], [18, 9, 16, 8]])
(28, 56, [[14, 14, 1, 1], [14, 14, 2, 2], [14, 14, 3, 3], [4, 24, 1, 6], [8, 20, 2, 5], [12, 16, 3, 4], [16, 12, 4, 3], [20, 8, 5, 2], [24, 4, 6, 1], [7, 21, 2, 6], [14, 14, 4, 4], [21, 7, 6, 2], [14, 14, 5, 5], [14, 14, 6, 6], [2, 26, 1, 13], [4, 24, 2, 12], [6, 22, 3, 11], [8, 20, 4, 10], [10, 18, 5, 9], [12, 16, 6, 8], [14, 14, 7, 7], [16, 12, 8, 6], [18, 10, 9, 5], [20, 8, 10, 4], [22, 6, 11, 3], [24, 4, 12, 2], [26, 2, 13, 1], [7, 21, 4, 12], [14, 14, 8, 8], [21, 7, 12, 4], [14, 14, 9, 9], [14, 14, 10, 10], [4, 24, 3, 18], [8, 20, 6, 15], [12, 16, 9, 12], [16, 12, 12, 9], [20, 8, 15, 6], [24, 4, 18, 3], [14, 14, 11, 11], [7, 21, 6, 18], [14, 14, 12, 12], [21, 7, 18, 6], [14, 14, 13, 13], [2, 26, 2, 26], [4, 24, 4, 24], [6, 22, 6, 22], [8, 20, 8, 20], [10, 18, 10, 18], [12, 16, 12, 16], [14, 14, 14, 14], [16, 12, 16, 12], [18, 10, 18, 10], [20, 8, 20, 8], [22, 6, 22, 6], [24, 4, 24, 4], [26, 2, 26, 2]])
(29, 0, [])
(30, 75, [[10, 20, 1, 2], [20, 10, 2, 1], [15, 15, 2, 2], [6, 24, 1, 4], [12, 18, 2, 3], [18, 12, 3, 2], [24, 6, 4, 1], [10, 20, 2, 4], [20, 10, 4, 2], [15, 15, 4, 4], [10, 20, 3, 6], [20, 10, 6, 3], [6, 24, 2, 8], [12, 18, 4, 6], [18, 12, 6, 4], [24, 6, 8, 2], [5, 25, 2, 10], [10, 20, 4, 8], [15, 15, 6, 6], [20, 10, 8, 4], [25, 5, 10, 2], [2, 28, 1, 14], [4, 26, 2, 13], [6, 24, 3, 12], [8, 22, 4, 11], [10, 20, 5, 10], [12, 18, 6, 9], [14, 16, 7, 8], [16, 14, 8, 7], [18, 12, 9, 6], [20, 10, 10, 5], [22, 8, 11, 4], [24, 6, 12, 3], [26, 4, 13, 2], [28, 2, 14, 1], [15, 15, 8, 8], [10, 20, 6, 12], [20, 10, 12, 6], [3, 27, 2, 18], [6, 24, 4, 16], [9, 21, 6, 14], [12, 18, 8, 12], [15, 15, 10, 10], [18, 12, 12, 8], [21, 9, 14, 6], [24, 6, 16, 4], [27, 3, 18, 2], [10, 20, 7, 14], [20, 10, 14, 7], [5, 25, 4, 20], [10, 20, 8, 16], [15, 15, 12, 12], [20, 10, 16, 8], [25, 5, 20, 4], [6, 24, 5, 20], [12, 18, 10, 15], [18, 12, 15, 10], [24, 6, 20, 5], [10, 20, 9, 18], [20, 10, 18, 9], [15, 15, 14, 14], [2, 28, 2, 28], [4, 26, 4, 26], [6, 24, 6, 24], [8, 22, 8, 22], [10, 20, 10, 20], [12, 18, 12, 18], [14, 16, 14, 16], [16, 14, 16, 14], [18, 12, 18, 12], [20, 10, 20, 10], [22, 8, 22, 8], [24, 6, 24, 6], [26, 4, 26, 4], [28, 2, 28, 2]])

northwolves

显然$a(p)=0$