HDU 2256 & HDU 4565 （矩阵快速幂 + 公式推演）

HDU 2256

题意：

计算⌊(2√+3√)2n⌋mod1024\lfloor{(\sqrt2 +\sqrt3)^{2n}}\rfloor \mod1024

思路：

∵f(n)=(2√+3√)2n=(5+26√)n=An+Bn∗6√\because f(n)={(\sqrt2 +\sqrt3)^{2n}} = {(5 +2\sqrt6)^{n}}= A_n+B_n*\sqrt6

∴f(n−1)=An−1+Bn−1∗6√\therefore f(n-1)= A_{n-1}+B_{n-1}*\sqrt6

又∵f(n)=(5+26√)∗f(n−1)又\because f(n)= (5+2\sqrt6)*f(n-1)

∴f(n)=(5∗An−1+12∗Bn−1)+(2∗An−1+5∗Bn−1)∗6√\therefore f(n)= (5*A_{n-1}+12*B_{n-1})+(2*A_{n-1}+5*B_{n-1})*\sqrt6

所以递推矩阵就是：

(52125)∗(An−1Bn−1)=(AnBn)\left(
\begin{array}{cc}
5 & 12 \\
2 & 5
\end{array}
\right)
*
\left(
\begin{array}{cc}
A_{n-1}\\
B_{n-1}
\end{array}
\right)
=\left(
\begin{array}{cc}
A_{n}\\
B_{n}
\end{array}
\right)


A1=5,B1=2.A_1=5,B_1=2.

然后套矩阵快速幂模板即可求出An,BnA_n,B_n.

又∵(5+26√)n=An+Bn∗6√又\because (5+2\sqrt6)^n = A_n+B_n*\sqrt6

∴(5−26√)n=An−Bn∗6√\therefore (5-2\sqrt6)^n = A_n-B_n*\sqrt6

∴(5+26√)n+(5−26√)n=2An\therefore (5+2\sqrt6)^n +(5-2\sqrt6)^n = 2A_n

又∵(5−26√)n<1又\because (5-2\sqrt6)^n <1

∴⌊(5+26√)n⌋=2An−1\therefore \lfloor(5+2\sqrt6)^n\rfloor = 2A_n-1

所以最后答案就是2An−12A_n-1

代码：

[code]/*
* @author FreeWifi_novicer
* language : C++/C
*/
#include<cstdio>
#include<iostream>
#include<cstring>
#include<cstdlib>
#include<cmath>
#include<algorithm>
#include<string>
#include<map>
#include<set>
#include<vector>
#include<queue>

using namespace std;

#define clr( x , y ) memset(x,y,sizeof(x))
#define cls( x ) memset(x,0,sizeof(x))
#define mp make_pair
#define pb push_back
typedef long long lint;
typedef long long ll;
typedef long long LL;

const int mod = 1024;
const int maxn = 4;
struct Matrix{
    int n,m;
    lint a[maxn][maxn];
    Matrix(int n , int m){
        this->n = n;
        this->m = m;
        cls(a);
    }
    Matrix operator * (const Matrix &tmp){
        Matrix res(n,tmp.m);
        for(int i = 0 ; i < n ; i++){
            for(int j = 0 ; j < tmp.m ; j++){
                for(int k = 0 ; k < m ; k++){
                    res.a[i][j] = (res.a[i][j] + (a[i][k] * tmp.a[k][j]) % mod) % mod;
                }
            }
        }
        return res;
    }
};

void Matrix_print(Matrix x){
    for(int i = 0 ; i < x.n ; i++){
        for(int j = 0 ; j < x.m ; j++) cout << x.a[i][j] << ' ';
        cout << endl;
    }
}
Matrix fast_pow(Matrix x ,int n){
    Matrix res(x.n,x.m);
    for(int i = 0 ; i <  x.n ; i++) res.a[i][i] = 1;
    while(n){
        if(n&1)
            res = res * x;
        x = x*x;
        n >>= 1;
        //Matrix_print(res);
        //cout << endl;
    }
    return res;
}

void solve(){
    lint n;
    cin >> n;
    if(n == 1){
        cout << 9 << endl;
        return;
    }
    Matrix base(2,1);
    Matrix fun(2,2);
    base.a[0][0] = 5;
    base.a[1][0] = 2;
    fun.a[0][0] = 5;
    fun.a[0][1] = 12;
    fun.a[1][0] = 2;
    fun.a[1][1] = 5;
    fun = fast_pow(fun,n-1);
    base = fun * base;
    cout << (2*base.a[0][0] - 1) % mod << endl;
}

int main(){
    int t ; cin >> t;
    while(t--){
        solve();
    }
    return 0;
}

附送HDU 4565代码:

[code]/*
* @author FreeWifi_novicer
* language : C++/C
*/
#include<cstdio>
#include<iostream>
#include<cstring>
#include<cstdlib>
#include<cmath>
#include<algorithm>
#include<string>
#include<map>
#include<set>
#include<vector>
#include<queue>

using namespace std;

#define clr( x , y ) memset(x,y,sizeof(x))
#define cls( x ) memset(x,0,sizeof(x))
#define mp make_pair
#define pb push_back
typedef long long lint;
typedef long long ll;
typedef long long LL;
lint mod;
lint n , a , b ;
const int maxn = 4;
struct Matrix{
    int n,m;
    lint a[maxn][maxn];
    Matrix(int n , int m){
        this->n = n;
        this->m = m;
        cls(a);
    }
    Matrix operator * (const Matrix &tmp){
        Matrix res(n,tmp.m);
        for(int i = 0 ; i < n ; i++){
            for(int j = 0 ; j < tmp.m ; j++){
                for(int k = 0 ; k < m ; k++){
                    res.a[i][j] = (res.a[i][j] + (a[i][k] * tmp.a[k][j]) % mod) % mod;
                }
            }
        }
        return res;
    }
};

void Matrix_print(Matrix x){
    for(int i = 0 ; i < x.n ; i++){
        for(int j = 0 ; j < x.m ; j++) cout << x.a[i][j] << ' ';
        cout << endl;
    }
}
Matrix fast_pow(Matrix x ,int n){
    Matrix res(x.n,x.m);
    for(int i = 0 ; i <  x.n ; i++) res.a[i][i] = 1;
    while(n){
        if(n&1)
            res = res * x;
        x = x*x;
        n >>= 1;
        //Matrix_print(res);
        //cout << endl;
    }
    return res;
}

void solve(){
    if(n == 1){
        lint ans = (lint)( a + ceil( sqrt( b * 1.0 ) ) ) % mod;
        cout << ans << endl;
        return;
    }
    Matrix base(2,1);
    Matrix fun(2,2);
    base.a[0][0] = a % mod;
    base.a[1][0] = 1;
    fun.a[0][0] = a % mod;
    fun.a[0][1] = b % mod;
    fun.a[1][0] = 1;
    fun.a[1][1] = a % mod;
    fun = fast_pow(fun,n-1);
    base = fun * base;
    cout << (2 * base.a[0][0]) % mod << endl;
}

int main(){
    while(cin >> a >> b >> n >> mod ){
        solve();
    }
    return 0;
}