【LOJ】#2535. 「CQOI2018」九连环

题解

简单分析一下，有\(k\)个环肯定是，我拆掉了\(k - 2\)个，留最左两个，1步拆掉最左的，这个时候我还要把这\(k - 2\)个环拼回去，拆一次\(k - 1\)

所以方案数就是\(f[k] = f[k - 1] + 2 * f[k - 2] + 1\)

然而太简单了，简单的都不是省选题了，所以他没让你取模= =，让你写FFT的高精乘，强行增加代码量

这个矩阵有几个位置默认是0，可以通过不对那里进行运算减小常数

代码

#include <bits/stdc++.h>
#define fi first
#define se second
#define pii pair<int, int>
#define pdi pair<db, int>
#define mp make_pair
#define pb push_back
#define enter putchar('\n')
#define space putchar(' ')
#define eps 1e-8
#define mo 974711
#define MAXN 1000005
//#define ivorysi
using namespace std;
typedef long long int64;
typedef double db;
template <class T>
void read(T &res) {
    res = 0;
    char c = getchar();
    T f = 1;
    while (c < '0' || c > '9') {
        if (c == '-') f = -1;
        c = getchar();
    }
    while (c >= '0' && c <= '9') {
        res = res * 10 + c - '0';
        c = getchar();
    }
    res *= f;
}
template <class T>
void out(T x) {
    if (x < 0) {
        x = -x;
        putchar('-');
    }
    if (x >= 10) {
        out(x / 10);
    }
    putchar('0' + x % 10);
}
const int BASE = 10;
const int len = 8;
const int MOD = 998244353, MAXL = (1 << 20);
int W[MAXL + 5];
int inc(int a, int b) { return a + b >= MOD ? a + b - MOD : a + b; }
int mul(int a, int b) { return 1LL * a * b % MOD; }
int fpow(int x, int c) {
    int res = 1, t = x;
    while (c) {
        if (c & 1) res = 1LL * res * t % MOD;
        t = 1LL * t * t % MOD;
        c >>= 1;
    }
    return res;
}
struct Bignum {
    vector<int> v;
    Bignum() { *this = 0; }
    Bignum operator=(int64 x) {
        v.clear();
        do {
            v.pb(x % BASE);
            x /= BASE;
        } while (x);
        return *this;
    }
    friend Bignum operator+(const Bignum &a, const Bignum &b) {
        Bignum c;
        c.v.clear();
        int p = 0, g = 0;
        while (1) {
            int x = g;
            if (p < a.v.size()) x += a.v[p];
            if (p < b.v.size()) x += b.v[p];
            if (!x && p >= a.v.size() && p >= b.v.size()) break;
            c.v.pb(x % BASE);
            g = x / BASE;
            ++p;
        }
        return c;
    }
    friend void NTT(Bignum &a, int L, int on) {
        a.v.resize(L, 0);
        for (int i = 1, j = L >> 1; i < L - 1; ++i) {
            if (i < j) swap(a.v[i], a.v[j]);
            int k = L >> 1;
            while (j >= k) {
                j -= k;
                k >>= 1;
            }
            j += k;
        }
        for (int h = 2; h <= L; h <<= 1) {
            int wn = W[(MAXL + MAXL * on / h) % MAXL];
            for (int k = 0; k < L; k += h) {
                int w = 1;
                for (int j = k; j < k + h / 2; ++j) {
                    int u = a.v[j], t = mul(w, a.v[j + h / 2]);
                    a.v[j] = inc(u, t);
                    a.v[j + h / 2] = inc(u, MOD - t);
                    w = mul(w, wn);
                }
            }
        }
        if (on == -1) {
            int InvL = fpow(L, MOD - 2);
            for (int i = 0; i < L; ++i) a.v[i] = mul(a.v[i], InvL);
        }
    }
    friend Bignum operator*(Bignum a, Bignum b) {
        Bignum c;
        int t = (a.v.size() + b.v.size()), L;
        L = 1;
        while (L <= t) L <<= 1;
        NTT(a, L, 1);
        NTT(b, L, 1);
        c.v.resize(L);
        for (int i = 0; i < L; ++i) c.v[i] = mul(a.v[i], b.v[i]);
        NTT(c, L, -1);
        int64 g = 0;
        for (int i = 0; i < L; ++i) {
            int64 x = g + c.v[i];
            c.v[i] = x % BASE;
            g = x / BASE;
        }
        while (g) {
            c.v.pb(g % BASE);
            g /= BASE;
        }
        L = c.v.size() - 1;
        while (L && c.v[L] == 0) {
            c.v.pop_back();
            --L;
        }
        return c;
    }
    void print() {
        for (int i = v.size() - 1; i >= 0; --i) {
            putchar('0' + v[i]);
        }
    }
} num;
struct Matrix {
    Bignum f[3][3];
    friend Matrix operator*(const Matrix &a, const Matrix &b) {
        Matrix c;
        c.f[2][2] = 1;
        for (int i = 0; i < 2; ++i) {
            for (int j = 0; j < 2; ++j) {
                for (int k = 0; k < 2; ++k) {
                    c.f[i][j] = c.f[i][j] + a.f[i][k] * b.f[k][j];
                }
            }
        }
        for (int j = 0; j < 2; ++j) {
            for (int k = 0; k < 3; ++k) {
                c.f[2][j] = c.f[2][j] + a.f[2][k] * b.f[k][j];
            }
        }
        return c;
    }
} A, ans;
void Init() {
    W[0] = 1;
    W[1] = fpow(3, (MOD - 1) / MAXL);
    for (int i = 2; i < MAXL; ++i) {
        W[i] = mul(W[i - 1], W[1]);
    }
    A.f[0][0] = 1;
    A.f[0][1] = 1;
    A.f[1][0] = 2;
    A.f[2][0] = 1;
    A.f[2][2] = 1;
}
void fpow(Matrix &res, int c) {
    Matrix t = A;
    res = A;
    --c;
    while (c) {
        if (c & 1) res = res * t;
        t = t * t;
        c >>= 1;
    }
}
void Solve() {
    int N;
    read(N);
    if (N == 1) {
        puts("1");
    } else if (N == 2) {
        puts("2");
    } else {
        fpow(ans, N - 2);
        num = ans.f[0][0] + ans.f[0][0] + ans.f[1][0] + ans.f[2][0];
        num.print();
        enter;
    }
}
int main() {
#ifdef ivorysi
    freopen("f1.in", "r", stdin);
#endif
    Init();
    int T;
    read(T);
    while (T--) Solve();
    return 0;
}