HDU-6363 bookshelf 莫比烏斯反演
bookshelf
Problem Description
Patrick Star bought a bookshelf, he named it ZYG !!
Patrick Star has N book .
The ZYG has K layers (count from 1 to K) and there is no limit on the capacity of each layer !
Now Patrick want to put all N books on ZYG :
Assume that the i-th layer has cnti(0≤cnti≤N) books finally.
Assume that f[i] is the i-th fibonacci number (f[0]=0,f[1]=1,f[2]=1,f[i]=f[i−2]+f[i−1]).
Define the stable value of i-th layers stablei=f[cnti].
Define the beauty value of i-th layers beautyi=2stablei−1.
Define the whole beauty value of ZYG score=gcd(beauty1,beauty2,…,beautyk)(Note: gcd(0,x)=x).
Patrick Star wants to know the expected value of score if Patrick choose a distribute method randomly !
Input
The first line contain a integer T (no morn than 10), the following is T test case, for each test case :
Each line contains contains three integer n,k
Output
For each test case, output the answer as a value of a rational number modulo .
Formally, it is guaranteed that under given constraints the probability is always a rational number pq (p and q are integer and coprime, q is positive), such that q is not divisible by 109+7. Output such integer a between 0 and
Sample Input
1 6 8
Sample Output
797202805
題意:把N本書放到K層的書架上,每一層的美麗值為,其中cnt是這一層書的數量,fib[x]為斐波那契數列,整個書架的美麗值為,問整個書架的美麗值的期望
題解:考試的時候因為本人實在是太菜了。對這個題目完全沒有一點思路,考完試以後聽了dls直播算是對這個題目有了一點思路程式碼很短,但是從學習各種姿勢到寫完用了一天,感覺一下子掌握了好多神奇的姿勢,開心。
經典定理:
那麼對於這個書架的美麗值可以做如下變形:
於是我們設