求倍三角数对

nyy · 发表于 2023-5-8 10:30:34

本帖最后由 nyy 于 2023-5-8 11:43 编辑

\[\left\{\left\{x\to \text{ConditionalExpression}\left[\frac{1}{12} \left(\sqrt{3} \left(4 \sqrt{3}+7\right)^{c_1}-\left(4 \sqrt{3}+7\right)^{c_1}-\sqrt{3} \left(7-4 \sqrt{3}\right)^{c_1}-\left(7-4 \sqrt{3}\right)^{c_1}+2\right),c_1\in \mathbb{Z}\land c_1\geq 1\right],
y\to \text{ConditionalExpression}\left[\frac{1}{24} \left(-2 \left(-\sqrt{3} \left(4 \sqrt{3}+7\right)^{c_1}+\left(4 \sqrt{3}+7\right)^{c_1}+\sqrt{3} \left(7-4 \sqrt{3}\right)^{c_1}+\left(7-4 \sqrt{3}\right)^{c_1}-2\right)-12 \left(\frac{1}{6} \left(\sqrt{3} \left(4 \sqrt{3}+7\right)^{c_1}-3 \left(4 \sqrt{3}+7\right)^{c_1}-\sqrt{3} \left(7-4 \sqrt{3}\right)^{c_1}-3 \left(7-4 \sqrt{3}\right)^{c_1}\right)+1\right)\right),c_1\in \mathbb{Z}\land c_1\geq 1\right]\right\}\right\}\]

所有的正整数解

\[x=\left\{\text{ConditionalExpression}\left[\frac{1}{12} \left(2-\left(\sqrt{3}+1\right) \left(7-4 \sqrt{3}\right)^{c_1}+\left(\sqrt{3}-1\right) \left(7+4 \sqrt{3}\right)^{c_1}\right),c_1\in \mathbb{Z}\land c_1\geq 1\right]\right\}\]

nyy · 发表于 2023-5-8 11:47:17

王守恩发表于 2023-5-7 16:13
求 2x(x+1)=y(y+1) 的所有正整数解

1,x={2, 14, 84, 492, 2870, 16730, 97512, 568344, 3312 ...

求出一部分解:
\[\begin{array}{ll}
x\to 1 & y\to 2 \\
x\to 12 & y\to 32 \\
x\to 165 & y\to 450 \\
x\to 2296 & y\to 6272 \\
x\to 31977 & y\to 87362 \\
x\to 445380 & y\to 1216800 \\
x\to 6203341 & y\to 16947842 \\
x\to 86401392 & y\to 236052992 \\
x\to 1203416145 & y\to 3287794050 \\
\end{array}\]

代码如下：

Clear["Global`*"];(*清除所有变量*)
ans=Solve[2x(x+y)==y*(y+1)&&x>0&&y>0&&x<10^10,{x,y},Integers,Method->Reduce]
Grid[ans,Alignment->Left](*列表显示*)
x/.ans

复制代码

只有x的解如下
{1, 12, 165, 2296, 31977, 445380, 6203341, 86401392, 1203416145}

只有y的解如下：
{2, 32, 450, 6272, 87362, 1216800, 16947842, 236052992, 3287794050}

nyy · 发表于 2023-5-8 11:49:57

本帖最后由 nyy 于 2023-5-8 11:52 编辑

nyy 发表于 2023-5-8 11:47
求出一部分解:
\[\begin{array}{ll}
x\to 1 & y\to 2 \\

x的搜索结果如下
https://oeis.org/A046174
A046174 Indices of pentagonal numbers which are also triangular.
既是五角数又是三角数的数。

y没有找到

五边形数
https://baike.baidu.com/item/%E4 ... /9459853?fr=aladdin

三角数
https://baike.baidu.com/item/%E4 ... /8515478?fr=aladdin

nyy · 发表于 2023-5-8 12:20:05

nyy 发表于 2023-5-8 11:47
求出一部分解:
\[\begin{array}{ll}
x\to 1 & y\to 2 \\

https://oeis.org/A006253
A006253 Number of perfect matchings (or domino tilings) in C_4 X P_n.
(Formerly M1926)

这个y是上面的数列中第2、4、6、8、10...项。

王守恩 · 发表于 2023-5-8 13:34:02

nyy 发表于 2023-5-8 12:20
https://oeis.org/A006253
A006253 Number of perfect matchings (or domino tilings) in C_4 X P_n.
...

求 2x(x+1)=y(y+1) 的所有正整数解

Solve[{2 x (x + 1) == y (y + 1), 10^4 > x}, {x, y}, PositiveIntegers]

{{x -> 2, y -> 3}, {x -> 14, y -> 20}, {x -> 84, y -> 119}, {x -> 492,y -> 696}, {x -> 2870, y -> 4059}}

1,x={2, 14, 84, 492, 2870, 16730, 97512, 568344, 3312554, 19306982, 112529340, 655869060, 3822685022, 22280241074, 129858761424, 756872327472, 4411375203410, 25711378892990,}
x(n)=LinearRecurrence[{7, -7, 1}, {0, 0, 2}, n]=(Fibonacci[2 n + 1, 2] - 1)/2

2,y={3, 20, 119, 696, 4059, 23660, 137903, 803760, 4684659, 27304196, 159140519, 927538920, 5406093003, 31509019100, 183648021599, 1070379110496, 6238626641379, 36361380737780,}
y(n)=LinearRecurrence[{7, -7, 1}, {-1, 0, 3}, n]=(LucasL[2*n + 1, 2] - 2)/4

x,y与下面这串数有关系(具体怎么扯，我也不知道)
Table[FromContinuedFraction[ContinuedFraction[Sqrt[2], n]], {n, 1, 20}]
{1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, 577/408, 1393/985, 3363/2378, 8119/5741, 19601/13860, 47321/33461, 114243/80782, 275807/195025, 665857/470832, 1607521/1136689,

譬如1: 2=1*2,14=2*7,84=7*12,492=12*41,2870=41*70,16730=70*239,97512=239*408,568344=408*1393,3312554=13932378,
譬如2: 3=1*3,20=2*2*5,119=7*17,696=2*12*29,4059=41*99,23660=2*70*169,137903=239*577,803760=2*408*985,4684659=1393*3363,

nyy · 发表于 2023-5-8 13:55:48

本帖最后由 nyy 于 2023-5-8 13:57 编辑

王守恩发表于 2023-5-8 13:34
求 2x(x+1)=y(y+1) 的所有正整数解

Solve[{2 x (x + 1) == y (y + 1), 10^4 > x}, {x, y}, Positive ...

你以为只有你会递推？？？？？？？？
我也给你来个递推！

Clear["Global`*"];(*清除所有变量*)
(*可以先用穷举法解出x、y的前几项*)
xx={1,12,165,2296,31977,445380,6203341,86401392,1203416145}
yy={2,32,450,6272,87362,1216800,16947842,236052992,3287794050}
(*对于坐标x,求解n的表达式,n是变量*)
xxx=FindSequenceFunction[xx,n]//FullSimplify
(*对于坐标y,求解n的表达式,n是变量*)
yyy=FindSequenceFunction[yy,n]//FullSimplify
(*求解递推系数*)
FindLinearRecurrence[xx]
FindLinearRecurrence[yy]

复制代码

求解结果如下
x表达式如下：
\[\frac{1}{12} \left(\left(\sqrt{3}-1\right) \left(4 \sqrt{3}+7\right)^n-\left(\sqrt{3}+1\right) \left(7-4 \sqrt{3}\right)^n+2\right)\]

y表达式如下：
\[\frac{1}{6} \left(\left(4 \sqrt{3}+7\right)^n+\left(7-4 \sqrt{3}\right)^n-2\right)\]

x与y的递推系数如下：
{15, -15, 1}
{15, -15, 1}
结果表示
a[n+1]=15*a[n]-15*a[n-1]+1*a[n-2]
应该就是这个表达式，我没有好好地区验证！

nyy · 发表于 2023-5-8 14:31:03

王守恩发表于 2023-5-8 13:34
求 2x(x+1)=y(y+1) 的所有正整数解

Solve[{2 x (x + 1) == y (y + 1), 10^4 > x}, {x, y}, Positive ...

Clear["Global`*"];(*清除所有变量*)
(*求出(x,y)的前几组数据*)
ans=Solve[2*x*(x+1)==y*(y+1)&&x>0&&y>0&&x<10^8,{x,y},Integers,Method->Reduce]//FullSimplify
Grid[ans,Alignment->Left](*列表显示*)
(*得到x、y的数列*)
xx=x/.ans
yy=y/.ans
(*对于坐标x,求解n的表达式,n是变量*)
xxx=FindSequenceFunction[xx,n]//FullSimplify
(*对于坐标y,求解n的表达式,n是变量*)
yyy=FindSequenceFunction[yy,n]//FullSimplify
(*求解递推系数*)
FindLinearRecurrence[xx]
FindLinearRecurrence[yy]

复制代码

前几组的解
\[\begin{array}{ll}
x\to 2 & y\to 3 \\
x\to 14 & y\to 20 \\
x\to 84 & y\to 119 \\
x\to 492 & y\to 696 \\
x\to 2870 & y\to 4059 \\
x\to 16730 & y\to 23660 \\
x\to 97512 & y\to 137903 \\
x\to 568344 & y\to 803760 \\
x\to 3312554 & y\to 4684659 \\
x\to 19306982 & y\to 27304196 \\
\end{array}\]

xx={2, 14, 84, 492, 2870, 16730, 97512, 568344, 3312554, 19306982}
yy={3, 20, 119, 696, 4059, 23660, 137903, 803760, 4684659, 27304196}

x的通项表达式
\[\frac{1}{8} \left(\left(7 \sqrt{2}+10\right) \left(2 \sqrt{2}+3\right)^{n-1}+\left(\sqrt{2}+2\right) \left(3-2 \sqrt{2}\right)^{n+1}-4\right)\]

y的通项表达式
\[\frac{1}{4} \left(\left(\sqrt{2}+1\right) \left(2 \sqrt{2}+3\right)^n-\left(\sqrt{2}-1\right) \left(3-2 \sqrt{2}\right)^n-2\right)\]

x与y的递推系数
{7, -7, 1}
{7, -7, 1}
a[n+1]=7*a[n]-7*a[n-1]+1*a[n-2]

nyy · 发表于 2023-5-8 14:37:04

nyy 发表于 2023-5-8 14:31
前几组的解
\[\begin{array}{ll}
x\to 2 & y\to 3 \\

x的数据
https://oeis.org/A053141

A053141 a(0)=0, a(1)=2 then a(n) = a(n-2) + 2*sqrt(8*a(n-1)^2 + 8*a(n-1) + 1).

y的数据
https://oeis.org/A001652
A001652 a(n) = 6*a(n-1) - a(n-2) + 2 with a(0) = 0, a(1) = 3

为什么OEIS上得到的数列，与我得到的数列的递推表达式不一样呢？

nyy · 发表于 2023-5-9 09:39:06

王守恩发表于 2023-5-8 13:34
求 2x(x+1)=y(y+1) 的所有正整数解

Solve[{2 x (x + 1) == y (y + 1), 10^4 > x}, {x, y}, Positive ...

譬如1: 2=1*2,14=2*7,84=7*12,492=12*41,2870=41*70,16730=70*239,97512=239*408,568344=408*1393,3312554=13932378,
譬如2: 3=1*3,20=2*2*5,119=7*17,696=2*12*29,4059=41*99,23660=2*70*169,137903=239*577,803760=2*408*985,4684659=1393*3363,
这个确实与根号2的连分数有关系！你挺能发现的！至少我发现不了！

王守恩 · 发表于 2023-5-9 12:21:38

nyy 发表于 2023-5-9 09:39
譬如1: 2=1*2,14=2*7,84=7*12,492=12*41,2870=41*70,16730=70*239,97512=239*408,568344=408*1393,331255 ...

Table[FromContinuedFraction[ContinuedFraction[Sqrt[2], 2 n]], {n, 1, 20}]

复制代码

\(1,\frac{3}{2},\frac{7}{5},\frac{17}{12},\frac{41}{29},\frac{99}{70},\frac{239}{169},\frac{577}{408},\frac{1393}{985},\frac{3363}{2378},\frac{8119}{5741},\frac{19601}{13860},\frac{47321}{33461},\frac{114243}{80782}, \frac{275807}{195205},
\frac{665857}{470832},\frac{1607521}{1136689},\frac{3880899}{2744210},\frac{9369319}{6625109},\frac{22619537}{15994428},\)

看分子:3=1*3, 20=3*7-1, 119=7*17, 696=17*41-1, 4059=41*99, 23660=99*239-1, 137903=239*577, 803760=577*1393-1, 4684659=1393*3363, 27304196,

看分母:2=(0+1)*2, 14=2*(2+5), 84=(2+5)*12, 492=12*(12+29), 2870=(12+29)*70, 16730=70*(70+169), 97512=(70+169)*408, 568344=408*(408+985), 3312554,

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[提问] 求倍三角数对

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