兔年快乐

王守恩

谢谢 northwolves！ 是这么一道题(与14楼有关联的):
\(\cot^{-1}{(n)}=\cot^{-1}(a)+\cot^{-1}(b),\ \ \ n=0,1,2,3,4,5,6...\)   a,b=正整数。
\(a(0)=1: \cot^{-1}{(0)}=\cot^{-1}{(1)}+\cot^{-1}{(1)}\)
\(a(1)=1: \cot^{-1}{(1)}=\cot^{-1}{(2)}+\cot^{-1}{(3)}\)
\(a(2)=1: \cot^{-1}{(2)}=\cot^{-1}{(3)}+\cot^{-1}{(7)}\)
\(a(3)=2: \cot^{-1}{(3)}=\cot^{-1}{(4)}+\cot^{-1}{(13)}=\cot^{-1}{(5)}+\cot^{-1}{(8)}\)
\(a(4)=1: \cot^{-1}{(4)}=\cot^{-1}{(5)}+\cot^{-1}{(21)}\)
\(a(5)=2: \cot^{-1}{(5)}=\cot^{-1}{(6)}+\cot^{-1}{(31)}=\cot^{-1}{(7)}+\cot^{-1}{(18)}\)
\(a(6)=1: \cot^{-1}{(6)}=\cot^{-1}{(7)}+\cot^{-1}{(43)}\)
\(a(7)=3: \cot^{-1}{(7)}=\cot^{-1}{(8)}+\cot^{-1}{(57)}=\cot^{-1}{(9)}+\cot^{-1}{(32)}=\cot^{-1}{(12)}+\cot^{-1}{(17)}\)
......
得到这样一串数:
{1,1,1,2,1,2,1,3,2,2,1,2,2,4,1,2,1,4,3,2,1,4,2,4,1,2,1,4,2,2,2,4,3,4,2,2,1,4,3,2,1,3,2,6,2,2,2,8,2,2,
2,2,2,4,1,4,1,8,2,2,2,2,2,4,2,2,1,4,4,2,3,2,4,8,1,4,2,4,2,2,2,4,3,8,1,2,2,4,2,4,1,4,2,6,1,2,2,4,4,6}

《数学研发论坛》网友 northwolves 给出通项公式如下。

1. Table[Length@Solve[a*b==1+n*(a+b)&&a>=b>n,{a,b},Integers],{n,0,99}]

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我就纳闷：怎么会这么简单！！！慢慢琢磨。还是谢谢 northwolves！

王守恩

(1): 1,0,1,0,2,0,3,0,5,0,8,0,13,0,21,0,34,......
(2): 1,0,0,1,0,0,2,0,0,3,0,0,5,0,0,8,0,0,13,0,0,21,0,0,34,......
(3): 1,0,0,0,1,0,0,0,2,0,0,0,3,0,0,0,5,0,0,0,8,0,0,0,13,0,0,0,21,0,0,0,34,......

Treenewbee

王守恩 发表于 2023-6-11 10:33
(1): 1,0,1,0,2,0,3,0,5,0,8,0,13,0,21,0,34,......
(2): 1,0,0,1,0,0,2,0,0,3,0,0,5,0,0,8,0,0,13,0,0,21 ...

1. Table[CoefficientList[Series[1/(1 - x^n - x^(2 n)), {x, 0, 20*n}],
   
2.   x], {n, 1, 4}]

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{{1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,10946},{1,0,1,0,2,0,3,0,5,0,8,0,13,0,21,0,34,0,55,0,89,0,144,0,233,0,377,0,610,0,987,0,1597,0,2584,0,4181,0,6765,0,10946},{1,0,0,1,0,0,2,0,0,3,0,0,5,0,0,8,0,0,13,0,0,21,0,0,34,0,0,55,0,0,89,0,0,144,0,0,233,0,0,377,0,0,610,0,0,987,0,0,1597,0,0,2584,0,0,4181,0,0,6765,0,0,10946},{1,0,0,0,1,0,0,0,2,0,0,0,3,0,0,0,5,0,0,0,8,0,0,0,13,0,0,0,21,0,0,0,34,0,0,0,55,0,0,0,89,0,0,0,144,0,0,0,233,0,0,0,377,0,0,0,610,0,0,0,987,0,0,0,1597,0,0,0,2584,0,0,0,4181,0,0,0,6765,0,0,0,10946}}

Treenewbee

王守恩 发表于 2023-6-11 10:33
(1): 1,0,1,0,2,0,3,0,5,0,8,0,13,0,21,0,34,......
(2): 1,0,0,1,0,0,2,0,0,3,0,0,5,0,0,8,0,0,13,0,0,21 ...

1. Table[CoefficientList[Series[1/(1 - x^n - x^(2 n)), {x, 0, 20*n}],
   
2.   x], {n, 1, 4}]

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{{1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,10946},{1,0,1,0,2,0,3,0,5,0,8,0,13,0,21,0,34,0,55,0,89,0,144,0,233,0,377,0,610,0,987,0,1597,0,2584,0,4181,0,6765,0,10946},{1,0,0,1,0,0,2,0,0,3,0,0,5,0,0,8,0,0,13,0,0,21,0,0,34,0,0,55,0,0,89,0,0,144,0,0,233,0,0,377,0,0,610,0,0,987,0,0,1597,0,0,2584,0,0,4181,0,0,6765,0,0,10946},{1,0,0,0,1,0,0,0,2,0,0,0,3,0,0,0,5,0,0,0,8,0,0,0,13,0,0,0,21,0,0,0,34,0,0,0,55,0,0,0,89,0,0,0,144,0,0,0,233,0,0,0,377,0,0,0,610,0,0,0,987,0,0,0,1597,0,0,0,2584,0,0,0,4181,0,0,0,6765,0,0,0,10946}}

王守恩

谢谢 Treenewbee！好方法！杀鸡焉用牛刀？谢谢 Treenewbee！

1,0,5,0,9,0,13,0,17,0,21,0,25,0,29,......
1,0,0,5,0,0,9,0,0,13,0,0,17,0,0,21,0,0,25,0,0,29,......
1,0,0,0,5,0,0,0,9,0,0,0,13,0,0,0,17,0,0,0,21,0,0,0,25,0,0,0,29,......
1,0,0,0,0,5,0,0,0,0,9,0,0,0,0,13,0,0,0,0,17,0,0,0,0,21,0,0,0,0,25,0,0,0,0,29,......

Treenewbee

王守恩 发表于 2023-6-12 09:44
谢谢 Treenewbee！好方法！杀鸡焉用牛刀？谢谢 Treenewbee！

1,0,5,0,9,0,13,0,17,0,21,0,25,0,29,..... ...

1. Table[CoefficientList[Series[(1 + 3 x^n)/(1 - x^n)^2, {x, 0, 10*n}],
   
2.   x], {n, 1, 5}]

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{{1,5,9,13,17,21,25,29,33,37,41},{1,0,5,0,9,0,13,0,17,0,21,0,25,0,29,0,33,0,37,0,41},{1,0,0,5,0,0,9,0,0,13,0,0,17,0,0,21,0,0,25,0,0,29,0,0,33,0,0,37,0,0,41},{1,0,0,0,5,0,0,0,9,0,0,0,13,0,0,0,17,0,0,0,21,0,0,0,25,0,0,0,29,0,0,0,33,0,0,0,37,0,0,0,41},{1,0,0,0,0,5,0,0,0,0,9,0,0,0,0,13,0,0,0,0,17,0,0,0,0,21,0,0,0,0,25,0,0,0,0,29,0,0,0,0,33,0,0,0,0,37,0,0,0,0,41}}

王守恩

Treenewbee 发表于 2023-6-12 11:42
{{1,5,9,13,17,21,25,29,33,37,41},{1,0,5,0,9,0,13,0,17,0,21,0,25,0,29,0,33,0,37,0,41},{1,0,0,5, ...

谢谢 Treenewbee！！！  我只会这简单的(数字还动不了)。

1. Table[k*Boole[IntegerQ[k/n]]/n, {n, 1, 5}, {k, n, 40}]

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{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, ...  
{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, 0, 15, 0, 16, 0, 17, 0, ...
{1, 0, 0, 2, 0, 0, 3, 0, 0, 4, 0, 0, 5, 0, 0, 6, 0, 0, 7, 0, 0, 8, 0, 0, 9, 0, 0, 10, 0, 0, 11, 0, 0, 12, 0, 0, 13, 0},
{1, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 4, 0, 0, 0, 5, 0, 0, 0, 6, 0, 0, 0, 7, 0, 0, 0, 8, 0, 0, 0, 9, 0, 0, 0, 10},
{1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 4, 0, 0, 0, 0, 5, 0, 0, 0, 0, 6, 0, 0, 0, 0, 7, 0, 0, 0, 0, 8}

ccmmjj

找规律数这种事，王守恩网友最擅长啦。有数学程式软件，会容易一点哈。

王守恩

1. Solve[{x^2 + x*y - y^2 == 1, 2000 > x}, {x, y}, PositiveIntegers]

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{{x -> 1, y -> 1}, {x -> 2, y -> 3}, {x -> 5, y -> 8}, {x -> 13, y -> 21}, {x -> 34, y -> 55},
{x -> 89, y -> 144}, {x -> 233, y -> 377}, {x -> 610, y -> 987}, {x -> 1597, y -> 2584}}

王守恩

Treenewbee 发表于 2023-6-11 12:59
{{1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,10946},{1,0,1,0,2,0,3,0,5 ...

简单的数字串，OEIS好像没有。

1,6,12,17,23,28,34,39,45,50,56,61,67,72,78,83,89,94,100,......