2020 BIT冬训-二分三分快速幂矩阵 G - Kolya and Tanya CodeForces - 584B

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Problem Description

Kolya loves putting gnomes at the circle table and giving them coins, and Tanya loves studying triplets of gnomes, sitting in the vertexes of an equilateral triangle.

More formally, there are 3n gnomes sitting in a circle. Each gnome can have from 1 to 3 coins. Let‘s number the places in the order they occur in the circle by numbers from 0 to 3n?-?1, let the gnome sitting on the i-th place have a_i coins. If there is an integer i (0?≤?i?<?n) such that a_i?+?a_i?+?n?+?a_i?+?2n?≠?6, then Tanya is satisfied.

Count the number of ways to choose a_i so that Tanya is satisfied. As there can be many ways of distributing coins, print the remainder of this number modulo 10⁹?+?7. Two ways, a and b, are considered distinct if there is index i (0?≤?i?<?3n), such that a_i?≠?b_i (that is, some gnome got different number of coins in these two ways).

Input

A single line contains number n (1?≤?n?≤?10⁵) — the number of the gnomes divided by three.

Output

Print a single number — the remainder of the number of variants of distributing coins that satisfy Tanya modulo 10⁹?+?7.

Examples

1

20

2

680

Note

20 ways for n?=?1 (gnome with index 0 sits on the top of the triangle, gnome 1 on the right vertex, gnome 2 on the left vertex):

本题是组合数学。

题意为有n个三角形。只要其中一个三角形的三个顶点值加起来不为6即可。求方案总数（每个顶点为1~3）。

对于每个点来说有3种可能。n个三角形即3³ⁿ=27ⁿ种。其中每个三角形有7种情况不行（即6个123和1个222），即7ⁿ种。

固总数为27ⁿ-7ⁿ种可能。

一开始以为是20ⁿ.但这是每一个三角形的顶点相加都满足条件的情况。实际上只要有一个满足即可。

结果可以用快速幂和(a*b)%c=(a%c*b%c)%c来做。

由于这题的时间卡的不是很死。快速幂就懒得写了直接取余做就行了qwq

注意要开long long，不然会溢出。

#include<cstdio>
#include<algorithm>
using namespace std;
int n;
long long ans=1,temp=1;
int main(){
    scanf("%d",&n);
    for(int i=0;i<3*n;i++){
        ans=(3*ans)%(1000000007);
    }
    for(int i=0;i<n;i++){
        temp=(7*temp)%(1000000007);
    }
    printf("%d",(ans-temp+1000000007)%(1000000007));
}

2020 BIT冬训-二分三分快速幂矩阵 G - Kolya and Tanya CodeForces - 584B

标签：alt   tab   put   图片   any   方案总数   sam   class   str   

原文地址：https://www.cnblogs.com/mikku39/p/14407145.html