CERC_2008_Tower(hdu 2971)_tower hdu-优快云博客

本文链接：https://blog.youkuaiyun.com/utoppia/article/details/8810784

本文探讨了一种解决特定数学序列求和问题的高效算法，并通过实例展示了如何利用矩阵运算来简化计算过程。文章详细介绍了序列定义、变换步骤及最终求和公式，同时指出使用正确数据类型的重要性以避免运行时错误。通过给出具体代码实现，读者可以深入理解并应用此算法于实际问题中。

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PRO IS HERE

由 a(n) = 2*a(2)*a(n-1) - a(n-2)

let p = 2*a(2);

==>a(n) =p*a(n-1) - a(n-2)

==>a(n)^2 = p^2*a(n-1)^2 - 2*p*a(n-1)*a(n-2) + a(n-2)^2................(1)

let s(n) =sum( a(i) );

s(n) = s(n-1) + a(n)^2 ................................................................... (2)

a(n)*a(n-1) = p*a(n-1)^2 - a(n-1)*a(n-2)...................(3)

so we have:

a(n-1)^2 0 , 1 , 0 , 0 a(n-2)^2

a(n)^2 = 1 , p^2, -2*p, 0 * a(n-1)^2

a(n)*a(n-1) 0 , p , -1 , 0 a(n-1)*a(n-2)

s(n-1) 0 , 1 , 0 , 1 s(n-2)

use matrix to do this.

Problem is sovle now.

Ps: At begin I got TLE then I chang the methed but then Igot WA for amount of time becauseI used lld instand of I64d.BE remmenber to use I64d in hdu oj.

MY code :

#include<cstdio>

#define FOR(i,a,b) for(int (i)=(a);(i)<=(b);(i)++)
#define nMax 10
#define LL long long
LL mod;

template<typename T>
struct Matrix{
	T mtix[nMax][nMax];
	int n;
	Matrix(int n=1):n(n) {
		FOR(i,0,n-1) FOR(j,0,n-1) mtix[i][j] = 0;
		FOR(i,0,n-1) mtix[i][i] = 1;
	}
	friend Matrix operator * (const Matrix& a,const Matrix& b) {
		Matrix c(a.n);
		FOR(i,0,a.n-1) FOR(j,0,a.n-1) {
			c.mtix[i][j] = 0;
			FOR(k,0,a.n-1) if(a.mtix[i][k] && b.mtix[k][j]){
			    c.mtix[i][j] += a.mtix[i][k] * b.mtix[k][j] ;
                c.mtix[i][j]%=mod;
			}
		}
		return c;
	}
	friend Matrix Exp(Matrix a,int n){
		Matrix c(a.n);
		Matrix b = a;
		while(n){
            if(n&1) c=c*b;
            n >>= 1;
            b = b*b;
		}
		return c;
	}
	void init(T p) {
		p%=mod;
		FOR(i,0,n-1) FOR(j,0,n-1) mtix[i][j] = 0 ;
		mtix[1][1] = p*p%mod;
		mtix[0][1] = mtix[1][0] = mtix[3][1] = mtix[3][3] = 1;
		mtix[1][2] = ((-2*p) % mod + mod) % mod;
		mtix[2][1] = p;
		mtix[2][2] = ((-1  ) % mod + mod) % mod;
	}
};

LL b[5];
int main(){
	//freopen("input.txt","r",stdin);
	int n;
	LL a;
	int t;
	scanf("%d",&t);
	while(t--){
		scanf("%I64d%d%I64d",&a,&n,&mod);
		a%=mod;
        if(mod == 1){
            printf("0\n");
            continue;
		}
		Matrix<LL>  s(4);
		s.init(a*2);
		b[1]=a*a%mod,b[0]=1LL,b[2]=a,b[3]=1;
		s = Exp( s ,n-1 );
		LL ans = 0LL;
		FOR(i,0,3) ans = (ans+s.mtix[3][i]*b[i]) % mod;
		printf("%I64d\n",ans);
	}

	return 0;
}