Codeforces Round #118 (Div. 2) C. Plant （找规律）

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本文介绍了一种计算经过n年后朝上指向的三角形植物数量的方法。通过建立数学模型，利用递推公式和快速幂运算，实现了高效求解。文章提供了完整的C++代码实现。

题目链接：点击打开链接

题意：问你第n个大三角形中有几个小三角形是朝上的。

思路：每个朝上的小三角形下一年会变成三个朝上一个朝下，每个向下的小三角下年会变成三个朝下一个朝上。

我们可以设第i年朝上的是ai个朝下的是bi个，因此可以得到递推式：ai = 3*ai-1 + bi-1,

bi = 3*b*-1 + ai-1, 可以看到这个式子很像，两者做差得到ai-bi = 2*(ai-1 - bi-1)，可得到

第i年两者之差的公式，又可知每年有4^i个三角，即ai + bi = 4^i, ai - bi = 2^n 两者作和

除2即为答案，快速幂即可解决。

代码:

#include<bits/stdc++.h>
using namespace std;
typedef long long ll;
const ll mod = 1e9+7;
ll n;

ll p(ll a, ll k)
{
    a %= mod;
    ll ans = 1;
    while(k)
    {
        if(k&1) ans = ans*a%mod;
        a = a*a%mod;
        k /= 2;
    }
    return ans;
}

int main(void)
{
    while(cin >> n)
    {
        if(!n) puts("1");
        else printf("%I64d\n", (p(2, n-1)+p(2, 2*n-1))%mod);
    }
    return 0;
}

Dwarfs have planted a very interesting plant, which is a triangle directed "upwards". This plant has an amusing feature. After one year a triangle plant directed "upwards" divides into four triangle plants: three of them will point "upwards" and one will point "downwards". After another year, each triangle plant divides into four triangle plants: three of them will be directed in the same direction as the parent plant, and one of them will be directed in the opposite direction. Then each year the process repeats. The figure below illustrates this process.

$\text{[math]}$

Help the dwarfs find out how many triangle plants that point "upwards" will be in n years.

Input

The first line contains a single integer n (0 ≤ n ≤ 1018) — the number of full years when the plant grew.

Please do not use the %lld specifier to read or write 64-bit integers in С++. It is preferred to use cin, cout streams or the %I64dspecifier.

Output

Print a single integer — the remainder of dividing the number of plants that will point "upwards" in n years by 1000000007 (109 + 7).

        Examples
      
          input
        
1

          output
        
3

          input
        
2

          output
        
10

Note

The first test sample corresponds to the second triangle on the figure in the statement. The second test sample corresponds to the third one.