兔年快乐

王守恩

简单才是最好！这是最小解。

有这样一串数:  \(a(n)=\lfloor\frac{2n}{3}\rfloor\)

1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, 12, 12, 13, 14, 14, 15, 16, 16, 17, 18, 18, 19, ...

\((2^0)^{2}+(2^0)^{2}=(2^3)^{1}\)
\((2^1)^{2}+(2^1)^{2}=(2^1)^{3}\)
\((2^4)^{2}+(2^2)^{4}=(2^3)^{3}\)
\((2^2)^{4}+(2^2)^{4}=(2^3)^{3}\)
\((2^1)^{4}+(2^1)^{4}=(2^1)^{5}\)
\((2^6)^{4}+(2^4)^{6}=(2^5)^{5}\)
\((2^4)^{6}+(2^4)^{6}=(2^5)^{5}\)
\((2^1)^{6}+(2^1)^{6}=(2^1)^{7}\)
\((2^8)^{6}+(2^6)^{8}=(2^7)^{7}\)
\((2^6)^{8}+(2^6)^{8}=(2^7)^{7}\)
\((2^1)^{8}+(2^1)^{8}=(2^1)^{9}\)

王守恩

胆怯的问：这漂亮的等式是怎么来的？谢谢！

\(\arctan\big(\frac{1}{1}\big)=\arctan\big(\frac{1}{2}\big)+\arctan\big(\frac{1}{3}\big)\)

\(\arctan\big(\frac{1}{3}\big)=\arctan\big(\frac{1}{5}\big)+\arctan\big(\frac{1}{8}\big)\)

\(\arctan\big(\frac{1}{8}\big)=\arctan\big(\frac{1}{13}\big)+\arctan\big(\frac{1}{21}\big)\)

\(\arctan\big(\frac{1}{21}\big)=\arctan\big(\frac{1}{34}\big)+\arctan\big(\frac{1}{55}\big)\)

\(\arctan\big(\frac{1}{55}\big)=\arctan\big(\frac{1}{89}\big)+\arctan\big(\frac{1}{144}\big)\)
......

王守恩

今年是兔年，恭祝大家兔年快乐！

F=1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765,

\(F(01)=001\)
\(F(02)=001\)
\(F(03)=002=001*1+001*1\)
\(F(04)=003=002*1+001*1\)
\(F(05)=005=003*1+002*1=002*2+001*1\)
\(F(06)=008=005*1+003*1=003*2+002*1\)
\(F(07)=013=008*1+005*1=005*2+003*1=003*3+002*2\)
\(F(08)=021=013*1+008*1=008*2+005*1=005*3+003*2\)
\(F(09)=034=021*1+013*1=013*2+008*1=008*3+005*2=005*5+03*3\)
\(F(10)=055=034*1+021*1=021*2+013*1=013*3+008*2=008*5+05*3\)
\(F(11)=089=055*1+034*1=034*2+021*1=021*3+013*2=013*5+08*3=08*8+05*5\)
\(F(12)=144=089*1+055*1=055*2+034*1=034*3+021*2=021*5+13*3=13*8+08*5\)
\(F(13)=233=144*1+089*1=089*2+055*1=055*3+034*2=034*5+21*3=21*8+13*5=13*13+08*8\)
\(F(14)=377=233*1+144*1=144*2+089*1=089*3+055*2=055*5+34*3=34*8+21*5=21*13+13*8\)
\(F(15)=610=377*1+233*1=233*2+144*1=144*3+089*2=089*5+55*3=55*8+34*5=34*13+21*8=21*21+13*13\)
\(F(16)=987=610*1+377*1=377*2+233*1=233*3+144*2=144*5+89*3=89*8+55*5=55*13+34*8=34*21+21*13\)

王守恩

胆怯的问：这漂亮的等式是怎么来的？谢谢！

\(\arctan\big(\frac{1}{1}\big)=\arctan\big(\frac{1}{2}\big)+\arctan\big(\frac{1}{3}\big)\)

\(\arctan\big(\frac{1}{3}\big)=\arctan\big(\frac{1}{5}\big)+\arctan\big(\frac{1}{8}\big)\)

\(\arctan\big(\frac{1}{8}\big)=\arctan\big(\frac{1}{13}\big)+\arctan\big(\frac{1}{21}\big)\)

\(\arctan\big(\frac{1}{21}\big)=\arctan\big(\frac{1}{34}\big)+\arctan\big(\frac{1}{55}\big)\)

\(\arctan\big(\frac{1}{55}\big)=\arctan\big(\frac{1}{89}\big)+\arctan\big(\frac{1}{144}\big)\)
......

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}]

复制代码

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}]

复制代码

{{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}]

复制代码

{{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}}