Yet Another Number Sequence CodeForces - 392C 矩阵快速幂

题目链接:点我

Everyone knows what the Fibonacci sequence is. This sequence can be defined by the recurrence relation:
F1 = 1, F2 = 2, Fi = Fi - 1 + Fi - 2 (i > 2).

We'll define a new number sequence Ai(k) by the formula:
Ai(k) = Fi × i^k (i ≥ 1).

In this problem, your task is to calculate the following sum: A1(k) + A2(k) + ... + An(k). The answer can be very large, so print it modulo 1000000007 (109 + 7).

Input

The first line contains two space-separated integers n, k (1 ≤ n ≤ 1e17; 1 ≤ k ≤ 40).

Output

Print a single integer — the sum of the first n elements of the sequence Ai(k) modulo 1000000007 (1e9 + 7).

Example

Input

1 1

Output

1

Input

4 1

Output

34

Input

5 2

Output

316

Input

7 4

Output

73825

题意:

如果提上公式所说:计算 A1(k) + A2(k) + … + An(k), Ai(k) = Fi × i^k.

思路:

矩阵快速幂.

首先我们应该知道,(n+1)k =(kk)nk +(kk−1) nk−1 + (kk−2)nk−2 + ….. +(k1) n +(k0) n0

我们令u(n+1,k) = (n+1)k * F(n+1), v(n+1,k) = (n+1)k * F(n)

根据斐波那契数列递推式:

u(n+1,k) = (n+1)k * F(n+1)

= (n+1)k * F(n) + (n+1)k * F(n-1)

= ∑ki=0 (ki) * ni * F(n) + ∑ki=0 (ki) * ni * F(n-1)

= ∑ki=0 (ki) * ni * F(n) + ∑ki=0 (ki) * v(n,i)

v(n+1,k) = (n+1)k * F(n)

= ∑ki=0 (ki) * ni * F(n)

= ∑ki=0 (ki) * u(n,i)

令Sn=A1(k) + A2(k) + ... + An(k)

所以 Sn+1=Sn+u(n,k),

那么我们构造一个(2K+3) * (2k+3)的矩阵即可,这个矩阵大家根据上面的递推式在纸上写写就出来了,

代码:

#include<cstdio>
#include<algorithm>
#include<cstring>
#include<iostream>
#include<cmath>
#include<queue>
using namespace std;

typedef long long LL;
const int mod = 1e9+7;
int c[45][45];
LL m;

struct mat{
LL a[85][85];
mat (){memset(a,0 ,sizeof(a)); }
mat operator *(const mat q){
mat c;
for(int i = 1; i <= m; ++i)
for(int k = 1; k <= m; ++k)
if(a[i][k])
for(int j = 1; j <= m; ++j){
c.a[i][j] += a[i][k] * q.a[k][j];
if (c.a[i][j] >= mod) c.a[i][j] %= mod;
}return c;
}
};

mat qpow(mat x, LL n){
mat ans;
for(int i = 1; i <= 84; ++i)
ans.a[i][i] = 1;
while(n){
if (n&1) ans = ans * x;
x = x * x;
n >>= 1;
}return ans;
}

void init(){
for(int i = 0; i <= 41; ++i)
c[i][0] = c[i][i] = 1;
for(int i = 1; i <= 41; ++i)
for(int j = 1; j < i; ++j)
c[i][j] = (c[i-1][j] + c[i-1][j-1]) % mod;
}

int main(){
LL n, k;
init();
scanf("%lld %lld", &n, &k);
mat ans;
m = 2 * k + 3;
ans.a[1][1] = 1;
for(int i = 2; i <= m; ++i){
ans.a[i][1] = 0;
ans.a[1][i] = c[k][(i-2)%(k+1)];
}for(int i = 2; i <= k+2; ++i)
for(int j = 2; j <= i;++j)
ans.a[i][j] = ans.a[i][j+k+1] = ans.a[i+k+1][j] =  c[i-2][j-2];
ans = qpow(ans,n-1);
LL sum = 0;
for(int i = 1; i <= m; ++i)
sum = (sum + ans.a[1][i]) % mod;
printf("%lld\n",sum);
return 0;
}