题解 P5175 【数列】
\[n\leq 10^{18}
\]
这摆明要用矩阵……
\[ans=\sum_{i=1}^na_i^2
\]
\[a_n=x\cdot a_{n-1}+y\cdot a_{n-2}\Rightarrow
\]
\[a_n^2=x^2a_{n-1}^2+2xy\cdot a_{n-1}a_{n-2}+y^2a_{n-2}^2
\]
所以我们可以知道
\[\color{red}{Once\;we\;kown\;the\;number\;of\;a_{n-1}\;,\;a_{n-2}\;and\;a_{n-1}\cdot a_{n-2},}
\]
\[\color{green}{we\;can\;get\;the\;number\;of\;a_{n}}
\]
所以我们自然而然地想到构造矩阵
设S(n)=ans(n);
\[\begin{bmatrix}S_n\\a_{n+1}^2\\a_{n+1}\cdot a_{n}\\a_{n}^2\end{bmatrix}
=
\begin{bmatrix}
1&\;1&\;0&\;0
\\
0 &\;x^2&\;2\cdot xy&\;y^2
\\
0&\;x&\;y&\;0
\\
0&\;1&\;0&\;0
\end{bmatrix}
\times
\begin{bmatrix}S_{n-1}\\a_{n}^2\\a_{n}\cdot a_{n-1}\\a_{n-1}^2\end{bmatrix}
\]
然后我们最初的矩阵是
\[\begin{bmatrix}S_{1}\\a_{2}^2\\a_{2}\cdot a_{1}\\a_{1}^2\end{bmatrix}
\]
所以答案就是初始矩阵乘以n-1次构造矩阵
#include<cstdio>
#define ll long long
const ll mod = 1e9 + 7;
struct matrix {
ll va[5][5];
int line, cross;
void Mem() {
for (int i = 1; i <= 4; i++) {
for (int j = 1; j <= 4; j++) {
va[i][j] = 0;
}
}
return ;
}
};
matrix operator *(const matrix &a, const matrix &b) {
matrix c;
c.Mem();
for (int i = 1; i <= a.line; i++) {
for (int k = 1; k <= a.cross; k++) {
for (int j = 1; j <= b.cross; j++) {
c.va[i][j] = (c.va[i][j] + a.va[i][k] * b.va[k][j] % mod) % mod;
}
}
}
c.line = a.line;
c.cross = b.cross;
return c;
}
int Starseven(void) {
int t;
read(t);
matrix ans, txt;
while(t--) {
ll n, a1, a2, x, y;
read(n);
read(a1);
read(a2);
read(x);
read(y);
if(n == 1) {
ll ott = a1 * a1 % mod;
write(ott);
puts("");
continue;
}
else if(n == 2) {
ll ott = a1 * a1 % mod + a2 * a2 % mod;
ott %= mod;
write(ott);
puts("");
continue;
}
ans.Mem();
txt.Mem();
ans.line = 4;
ans.cross = 1;
ans.va[1][1] = a1 * a1 % mod;
ans.va[2][1] = a2 * a2 % mod;
ans.va[3][1] = a2 * a1 % mod;
ans.va[4][1] = a1 * a1 % mod;
txt.line = 4;
txt.cross = 4;
txt.va[1][1] = 1ll;
txt.va[1][2] = 1ll;
txt.va[2][2] = x * x % mod;
txt.va[2][3] = 2ll * x % mod * y % mod;
txt.va[2][4] = y * y % mod;
txt.va[3][2] = x;
txt.va[3][3] = y;
txt.va[4][2] = 1ll;
n -= 1ll;
while(n) {
if(n & 1ll) ans = txt * ans;
n >>= 1ll;
txt = txt * txt;
}
write(ans.va[1][1]);
puts("");
}
return 0;
}
