题目:Fibonacci Check-up
链接:
分析:
1)二项式展开:$(x+1)^n = \sum^n_{k=0}{C^k_n * x^k}$
2)Fibonacci数列可以写为:$ \left[ \begin{array}{cc} 0 & 1 \\ 1 & 1 \end{array} \right]^n$的左下角项。
3)构造矩阵$ T = Fib+E = \left[ \begin{array}{cc} 0 & 1 \\ 1 & 1 \end{array} \right] + \left[ \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right] = \left[ \begin{array}{cc} 1 & 1 \\ 1 & 2 \end{array} \right]$。
#include <iostream> #include <cstring> #include <cstdio> using namespace std; typedef long long LL; int MOD; struct Matrix{ LL a[2][2]; void init(int f){ memset(a,0,sizeof a); if(f==-1)return; for(int i=0;i<2;++i)a[i][i]=1; } }; Matrix operator*(Matrix& A,Matrix& B){ Matrix C;C.init(-1); for(int i=0;i<2;++i) for(int j=0;j<2;++j) for(int k=0;k<2;++k){ C.a[i][j]+=A.a[i][k]*B.a[k][j]; C.a[i][j]%=MOD; } return C; } Matrix operator^(Matrix A,int n){ Matrix Rt;Rt.init(0); for(;n;n>>=1){ if(n&1)Rt=Rt*A; A=A*A; } return Rt; } int main(){ int n,Case;scanf("%d",&Case); Matrix A,T; T.a[0][0]=1;T.a[0][1]=1; T.a[1][0]=1;T.a[1][1]=2; for(;Case--;){ scanf("%d%d",&n,&MOD); A=T^n; LL ans=A.a[1][0]; printf("%lld\n",ans%MOD); } return 0; }
4)$\sum^n_{k=0}{C^k_n * f(k)} = f(2*n) $
5)证明:$\sum^n_{k=0}{C^k_n * f(k)} = \sum^n_{k=0}{C^k_n * { $ $ { ( \frac{1+\sqrt{5}}{2} )}^n +{ ( \frac{1-\sqrt{5}}{2} )}^n$