本帖最后由 northwolves 于 2023-8-3 18:06 编辑
王守恩 发表于 2023-8-3 17:51
我们这串数: 0, 4, 48, 1440, 65280, ..., 通项公式(A189849/2^n)与 A189849不大一样。
如果用这串数， ...
n对夫妻入住某酒店的n个标间，夫妻不得同屋:lol
(1),Table + n/Cos, \/2 > a > 0}, {a}], {n, 1, 9}] // ToRadicals // FullSimplify
{2 Sqrt, Sqrt, Sqrt, Sqrt,
Sqrt,Sqrt,Sqrt,5 Sqrt, Sqrt}
(1)可以有通项公式:
Table + 3 Power]}, {s}], {n, 1, 9}] // ToRadicals // FullSimplify
s -> 2 Sqrt, s -> Sqrt, s -> Sqrt, s -> Sqrt,
s -> Sqrt, s -> Sqrt, s -> Sqrt, s -> 5 Sqrt, s -> Sqrt}
(2),Table + n/Cos, \/2 > a > 0}, {a}], {n, 1, 9}] // ToRadicals // FullSimplify
{Sqrt,4 Sqrt,Root[-2197 - 465 #^2 - 39 #^4 + #^6&,2,0],2 Sqrt,{Root[-24389 - 177 #^2 - 87 #^4 + #^6&,2,0],
2 Sqrt,Sqrt,2 Sqrt,Sqrt}
(3), Table + n/Cos, \/2 > a > 0}, {a}], {n, 1, 9}] // ToRadicals // FullSimplify
{Sqrt,Root[-2197 - 465 #^2 - 39 #^4 + #^6&,2,0],6 Sqrt, Sqrt,Sqrt,
3 Sqrt, Sqrt, Sqrt, 3 Sqrt]}
......
(1)可以有通项公式; (2),(3),(4)...可以有通项公式吗？
本帖最后由 王守恩 于 2023-8-7 10:16 编辑
(1),(2),(3),(4)...可以有统一的通项公式。
若\(\pi/2 > a > 0,\frac{m}{\sin(a)}+\frac{n}{\cos(a)}\)的最小值=\(\D(m^{2/3}+n^{2/3})^{3/2}\)
本帖最后由 northwolves 于 2023-8-7 13:44 编辑
$f(a)=k/Sin + n/Cos$
$f'(a)=\frac{kTan}{Cos}- \frac{n}{TanSin}$
$f'(a)=0->Tan=(\frac{k}{n})^{\frac{1}{3}}$
此时 $k/Sin + n/Cos=(k+nTan)\sqrt{1+\frac{1}{Tan^2)}=(k^{\frac{2}{3}}+n^{\frac{2}{3}})^{\frac{3}{2}}$
有这样一串数:4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, 216, 225,...A001597
1,每个数-1，取倒数相加，和是多少？1/3+1/7+1/8+1/15+1/24+1/26+1/31+1/35+1/48+1/63+1/80+1/99+1/120+...
2,每个数-指数(较小者)，取倒数相加，和是多少？1/2+1/5+1/7+1/14+1/23+24+1/27+1/34+1/47+1/62+1/79+1/98+...
$\sum_{k=2}^{∞}\frac{1}{n^k}=\frac{1}{n-1}-\frac{1}{n}$
$1/3+1/7+1/8+1/15+1/24+1/26+1/31+1/35+1/48+1/63+1/80+1/99+1/120+...=1-\frac{1}{2}+\frac1 2-\frac1 3+...=1$
N/Log]}],#<10^8&]],20]
1.4085485959754331397
第2题猜测极限值$\sqrt2$
一个数列的通项式，据说难倒英雄汉！
每一行的和都是"兔子数列"。每一列都是"杨辉三角"中的列。
Cn:Table[(n+k)!/((2k)!(n-k)!),{n,0,9},{k,0,n}]//TableForm
{1},
{1, 1},
{1, 3, 1},
{1, 6, 5, 1},
{1, 10, 15, 7, 1},
{1, 15, 35, 28, 9, 1},
{1, 21, 70, 84, 45, 11, 1},
{1, 28, 126, 210, 165, 66, 13, 1},
{1, 36, 210, 462, 495, 286, 91, 15, 1},
{1, 45, 330, 924, 1287, 1001, 455, 120, 17, 1}
An:Table[(n+k)!/((2k)!(n-k)!),{n,1,9},{k,0,n}]//TableForm
{1, 1},
{1, 3, 1},
{1, 6, 5, 1},
{1, 10, 15, 7, 1},
{1, 15, 35, 28, 9, 1},
{1, 21, 70, 84, 45, 11, 1},
{1, 28, 126, 210, 165, 66, 13, 1},
{1, 36, 210, 462, 495, 286, 91, 15, 1},
{1, 45, 330, 924, 1287, 1001, 455, 120, 17, 1}}]
Dn:Table[(n+k+1)!/((2k1)!(n-k)!),{n,0,9},{k,0,n}]//TableForm
{1},
{2, 1},
{3, 4, 1},
{4, 10, 6, 1},
{5, 20, 21, 8, 1},
{6, 35, 56, 36, 10, 1},
{7, 56, 126, 120, 55, 12, 1},
{8, 84, 252, 330, 220, 78, 14, 1},
{9, 120, 462, 792, 715, 364, 105, 16, 1},
{10,165, 792, 1716, 2002, 1365, 560, 136, 18, 1}}]
Bn:Table[(n+k+1)!/((2k+1)!(n-k)!),{n,1,9},{k,0,n}]//TableForm
{2, 1},
{3, 4, 1},
{4, 10, 6, 1},
{5, 20, 21, 8, 1},
{6, 35, 56, 36, 10, 1},
{7, 56, 126, 120, 55, 12, 1},
{8, 84, 252, 330, 220, 78, 14, 1},
{9, 120, 462, 792, 715, 364, 105, 16, 1},
{10,165, 792, 1716, 2002, 1365, 560, 136, 18, 1}
用足够多的a,b,c凑出n,一共有多少种方法？
CoefficientList],x]