線形漸化式用
(fps/linearly-recurrent-sequence.hpp)

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• Last update: 2025-10-21 21:13:36+09:00
• Include: #include "fps/linearly-recurrent-sequence.hpp"

線形漸化式に関するライブラリ．

• a, c を入力に含むとき，初項が $a_0,\dots,a_{d-1}$ で漸化式 $a_i=\sum_{j=1}^{d}c_{j-1}a_{i-j}$ で定まる数列．
• a のみを入力に含むとき，Berlekamp-Massey アルゴリズムで漸化式を推定した上で計算する．
• n, k を入力に含むとき，$a_n,a_{n+1},\dots,a_{n+k-1}$ を計算する．
• n のみを入力に含むとき，$a_n$ のみを計算する．

連続する項

https://noshi91.hatenablog.com/entry/2023/06/04/233447

$\deg P(x)\lt d,\deg Q(x)=d,[x^0]Q(x)\neq 0$ とする．

次を満たす $S(x)$ を求められればよい． \(\frac{P(x)}{Q(x)}=R(x)+x^n\cdot\frac{S(x)}{Q(x)},\quad \deg R(x)\lt n,\deg S(x)\lt d\)

$P(x)=Q(x)R(x)+x^nS(x)$ であるから $S(x)=x^{-n}P(x)\bmod Q(x)$ と表せる．

$\frac{1}{Q(x)}$ の $x^n$ から $x^{n+d-1}$ の係数を求めれば $P(x)$ のときの $S(x)$，すなわち $x^{-n}\bmod Q(x)$ が求められる．

inverse-shift.hpp で実装．

実質的に $\deg P(x)\geq\deg Q(x)$ であるようなケースに注意する．

例えば $a=(0,0,1,1,1,1,\dots)$ のようなケースに Berlekamp-Massey を用いると $q=(1,-1,0,0)$ が得られる．

このような場合，$q$ の末尾の $0$ の個数分 $a$ の先頭を取り除けばよい．

Depends on

• Berlekamp-Massey (fps/berlekamp-massey.hpp)
• Bostan-Mori (fps/bostan-mori.hpp)
• fps/formal-power-series.hpp
• Inverse の次数シフト (fps/inverse-shift.hpp)

Verified with

• verify/fps/LC_consecutive_terms_of_linear_recurrent_sequence.test.cpp
• verify/fps/LC_kth_term_of_linearly_recurrent_sequence.test.cpp

Code

#pragma
#include "fps/formal-power-series.hpp"
#include "fps/berlekamp-massey.hpp"
#include "fps/bostan-mori.hpp"
#include "fps/inverse-shift.hpp"

// a[i]=sum[j=1~d]c[j]a[i-j], i>=d
// find a[n]
template <class mint>
mint LinearyRecurrentSequence(FormalPowerSeries<mint> a, FormalPowerSeries<mint> c, long long n) {
  assert(a.size() == c.size());
  if (n < a.size()) return a[n];
  while (!c.empty() && c.back() == 0) c.pop_back();
  if (c.size() < a.size()) {
    int z = a.size() - c.size();
    n -= z;
    a.erase(a.begin(), a.begin() + z);
  }
  int d = c.size();
  FormalPowerSeries<mint> q(d + 1);
  q[0] = 1;
  for (int i = 0; i < d; i++) q[i + 1] = -c[i];
  auto p = (a * q).pre(d);
  return BostanMori(p, q, n);
}

// a[i]=sum[j=1~d]c[j]a[i-j], i>=d
// find a[n],a[n+1],...,a[n+k-1]
template <class mint>
FormalPowerSeries<mint> LinearyRecurrentSequence(FormalPowerSeries<mint> a, FormalPowerSeries<mint> c, long long n, int k) {
  assert(a.size() == c.size());
  if (n + k < a.size()) {
    a >>= (int)n;
    return a.pre(k);
  }
  while (!c.empty() && c.back() == 0) c.pop_back();
  int d = c.size();
  FormalPowerSeries<mint> ret{};
  if (c.size() < a.size()) {
    int z = a.size() - c.size();
    if (n < z) {
      ret.reserve(k);
      for (int i = n; i < z; i++) ret.push_back(a[i]);
    }
    n -= z;
    if (n < 0) {
      k += n;
      n = 0;
    }
    a >>= z;
  }
  FormalPowerSeries<mint> q(d + 1);
  q[0] = 1;
  for (int i = 0; i < d; i++) q[i + 1] = -c[i];
  auto p = (a * q).pre(d);
  if (n < a.size()) {
    p *= q.inv(n + k);
    p >>= n;
  } else {
    p *= (InverseShift(q, n) * q).pre(d);
    p %= q;
    p *= q.inv(k);
  }
  p.resize(k);
  if (ret.empty()) {
    return p;
  } else {
    for (auto v : p) ret.push_back(v);
    return ret;
  }
}

template <class mint>
mint LinearyRecurrentSequence(FormalPowerSeries<mint> a, long long n) {
  if (n < a.size()) return a[n];
  auto b = BerlekampMassey(a);
  int d = b.size() - 1;
  a.resize(d);
  int z = 0;
  while (b.back() == 0) b.pop_back(), z++;
  a >>= z;
  n -= z;
  d -= z;
  return BostanMori((a * b).pre(b.size()), b, n);
}

/**
 * @brief 線形漸化式用
 * @docs docs/fps/linearly-recurrent-sequence.md
 */

#line 1 "fps/linearly-recurrent-sequence.hpp"
#pragma
#line 2 "fps/formal-power-series.hpp"

template <class mint>
struct FormalPowerSeries : vector<mint> {
  using vector<mint>::vector;
  using FPS = FormalPowerSeries;
  FPS &operator+=(const FPS &r) {
    if (r.size() > this->size()) this->resize(r.size());
    for (int i = 0; i < (int)r.size(); i++) (*this)[i] += r[i];
    return *this;
  }
  FPS &operator+=(const mint &r) {
    if (this->empty()) this->resize(1);
    (*this)[0] += r;
    return *this;
  }
  FPS &operator-=(const FPS &r) {
    if (r.size() > this->size()) this->resize(r.size());
    for (int i = 0; i < (int)r.size(); i++) (*this)[i] -= r[i];
    return *this;
  }
  FPS &operator-=(const mint &r) {
    if (this->empty()) this->resize(1);
    (*this)[0] -= r;
    return *this;
  }
  FPS &operator*=(const mint &v) {
    for (int k = 0; k < (int)this->size(); k++) (*this)[k] *= v;
    return *this;
  }
  FPS &operator/=(const FPS &r) {
    if (this->size() < r.size()) {
      this->clear();
      return *this;
    }
    int n = this->size() - r.size() + 1;
    if ((int)r.size() <= 64) {
      FPS f(*this), g(r);
      g.shrink();
      mint coeff = g.at(g.size() - 1).inv();
      for (auto &x : g) x *= coeff;
      int deg = (int)f.size() - (int)g.size() + 1;
      int gs = g.size();
      FPS quo(deg);
      for (int i = deg - 1; i >= 0; i--) {
        quo[i] = f[i + gs - 1];
        for (int j = 0; j < gs; j++) f[i + j] -= quo[i] * g[j];
      }
      *this = quo * coeff;
      this->resize(n, mint(0));
      return *this;
    }
    return *this = ((*this).rev().pre(n) * r.rev().inv(n)).pre(n).rev();
  }
  FPS &operator%=(const FPS &r) {
    *this -= *this / r * r;
    shrink();
    return *this;
  }
  FPS operator+(const FPS &r) const { return FPS(*this) += r; }
  FPS operator+(const mint &v) const { return FPS(*this) += v; }
  FPS operator-(const FPS &r) const { return FPS(*this) -= r; }
  FPS operator-(const mint &v) const { return FPS(*this) -= v; }
  FPS operator*(const FPS &r) const { return FPS(*this) *= r; }
  FPS operator*(const mint &v) const { return FPS(*this) *= v; }
  FPS operator/(const FPS &r) const { return FPS(*this) /= r; }
  FPS operator%(const FPS &r) const { return FPS(*this) %= r; }
  FPS operator-() const {
    FPS ret(this->size());
    for (int i = 0; i < (int)this->size(); i++) ret[i] = -(*this)[i];
    return ret;
  }
  void shrink() {
    while (this->size() && this->back() == mint(0)) this->pop_back();
  }
  FPS rev() const {
    FPS ret(*this);
    reverse(begin(ret), end(ret));
    return ret;
  }
  FPS dot(FPS r) const {
    FPS ret(min(this->size(), r.size()));
    for (int i = 0; i < (int)ret.size(); i++) ret[i] = (*this)[i] * r[i];
    return ret;
  }
  FPS pre(int sz) const {
    return FPS(begin(*this), begin(*this) + min((int)this->size(), sz));
  }
  FPS operator>>=(int sz) {
    assert(sz >= 0);
    if ((int)this->size() <= sz) return {};
    this->erase(this->begin(), this->begin() + sz);
    return *this;
  }
  FPS operator>>(int sz) const {
    if ((int)this->size() <= sz) return {};
    FPS ret(*this);
    ret.erase(ret.begin(), ret.begin() + sz);
    return ret;
  }
  FPS operator<<=(int sz) {
    assert(sz >= 0);
    this->insert(this->begin(), sz, mint(0));
    return *this;
  }
  FPS operator<<(int sz) const {
    FPS ret(*this);
    ret.insert(ret.begin(), sz, mint(0));
    return ret;
  }
  FPS diff() const {
    const int n = (int)this->size();
    FPS ret(max(0, n - 1));
    mint one(1), coeff(1);
    for (int i = 1; i < n; i++) {
      ret[i - 1] = (*this)[i] * coeff;
      coeff += one;
    }
    return ret;
  }
  FPS integral() const {
    const int n = (int)this->size();
    FPS ret(n + 1);
    ret[0] = mint(0);
    if (n > 0) ret[1] = mint(1);
    auto mod = mint::get_mod();
    for (int i = 2; i <= n; i++) ret[i] = (-ret[mod % i]) * (mod / i);
    for (int i = 0; i < n; i++) ret[i + 1] *= (*this)[i];
    return ret;
  }
  mint eval(mint x) const {
    mint r = 0, w = 1;
    for (auto &v : *this) r += w * v, w *= x;
    return r;
  }
  FPS log(int deg = -1) const {
    assert((*this)[0] == mint(1));
    if (deg == -1) deg = (int)this->size();
    return (this->diff() * this->inv(deg)).pre(deg - 1).integral();
  }
  FPS pow(int64_t k, int deg = -1) const {
    const int n = (int)this->size();
    if (deg == -1) deg = n;
    if (k == 0) {
      FPS ret(deg);
      if (deg) ret[0] = 1;
      return ret;
    }
    for (int i = 0; i < n; i++) {
      if ((*this)[i] != mint(0)) {
        mint rev = mint(1) / (*this)[i];
        FPS ret = (((*this * rev) >> i).log(deg) * k).exp(deg);
        ret *= (*this)[i].pow(k);
        ret = (ret << (i * k)).pre(deg);
        if ((int)ret.size() < deg) ret.resize(deg, mint(0));
        return ret;
      }
      if (__int128_t(i + 1) * k >= deg) return FPS(deg, mint(0));
    }
    return FPS(deg, mint(0));
  }

  static void *ntt_ptr;
  static void set_ntt();
  FPS &operator*=(const FPS &r);
  FPS middle_product(const FPS &r) const;
  void ntt();
  void intt();
  void ntt_doubling();
  static int ntt_root();
  FPS inv(int deg = -1) const;
  FPS exp(int deg = -1) const;
};
template <typename mint>
void *FormalPowerSeries<mint>::ntt_ptr = nullptr;
#line 3 "fps/berlekamp-massey.hpp"

template <class mint>
FormalPowerSeries<mint> BerlekampMassey(const FormalPowerSeries<mint>& a) {
  int n = a.size();
  FormalPowerSeries<mint> b, c;
  b.reserve(n + 1), c.reserve(n + 1);
  b.push_back(1), c.push_back(1);
  mint y = 1;
  for (int k = 0; k < n; k++) {
    mint x = 0;
    for (int i = 0; i < c.size(); i++) x += c[i] * a[k - i];
    b.insert(b.begin(), 0);
    if (x == 0) continue;
    mint v = x / y;
    if (b.size() > c.size()) {
      for (int i = 0; i < b.size(); i++) b[i] *= -v;
      for (int i = 0; i < c.size(); i++) b[i] += c[i];
      swap(b, c);
      y = x;
    } else {
      for (int i = 0; i < b.size(); i++) c[i] -= v * b[i];
    }
  }
  return c;
}

/**
 * @brief Berlekamp-Massey
 * @docs docs/fps/berlekamp-massey.md
 */
#line 3 "fps/bostan-mori.hpp"

// [x^n]f(x)/g(x)
template <class mint>
mint BostanMori(FormalPowerSeries<mint> f, FormalPowerSeries<mint> g, long long n) {
  g.shrink();
  mint ret = 0;
  if (f.size() >= g.size()) {
    auto q = f / g;
    if (n < q.size()) ret += q[n];
    f -= q * g;
    f.shrink();
  }
  if (f.empty()) return ret;
  FormalPowerSeries<mint>::set_ntt();
  if (!FormalPowerSeries<mint>::ntt_ptr) {
    f.resize(g.size() - 1);
    for (; n > 0; n >>= 1) {
      auto g1 = g;
      for (int i = 1; i < g1.size(); i += 2) g1[i] = -g1[i];
      auto p = f * g1, q = g * g1;
      if (n & 1) {
        for (int i = 0; i < f.size(); i++) f[i] = p[(i << 1) | 1];
      } else {
        for (int i = 0; i < f.size(); i++) f[i] = p[i << 1];
      }
      for (int i = 0; i < g.size(); i++) g[i] = q[i << 1];
    }
    return ret + f[0] / g[0];
  } else {
    int m = 1, log = 0;
    while (m < g.size()) m <<= 1, log++;
    mint wi = mint(FormalPowerSeries<mint>::ntt_root()).inv().pow((mint::get_mod() - 1) >> (log + 1));
    vector<int> rev(m);
    for (int i = 0; i < rev.size(); i++) rev[i] = (rev[i / 2] / 2) | ((i & 1) << (log - 1));
    vector<mint> pow(m, 1);
    for (int i = 1; i < m; i++) pow[rev[i]] = pow[rev[i - 1]] * wi;
    f.resize(m), g.resize(m);
    f.ntt(), g.ntt();
    mint inv2 = mint(2).inv();
    for (; n >= m; n >>= 1) {
      f.ntt_doubling(), g.ntt_doubling();
      if (n & 1) {
        for (int i = 0; i < m; i++) f[i] = (f[i << 1] * g[(i << 1) | 1] - f[(i << 1) | 1] * g[i << 1]) * inv2 * pow[i];
      } else {
        for (int i = 0; i < m; i++) f[i] = (f[i << 1] * g[(i << 1) | 1] + f[(i << 1) | 1] * g[i << 1]) * inv2;
      }
      for (int i = 0; i < m; i++) g[i] = g[i << 1] * g[(i << 1) | 1];
      f.resize(m), g.resize(m);
    }
    f.intt(), g.intt();
    return ret + (f * g.inv())[n];
  }
}
/**
 * @brief Bostan-Mori
 * @docs docs/fps/bostan-mori.md
 */
#line 3 "fps/inverse-shift.hpp"

// [x^n,...,x^{n+k-1}]1/f(x)
template <class mint>
FormalPowerSeries<mint> InverseShift(FormalPowerSeries<mint> f, long long n, int k = -1) {
  using fps = FormalPowerSeries<mint>;
  assert(f[0] != 0);
  if (k == -1) k = f.size();
  int m = 1;
  while (m < k) m <<= 1;

  int log = __builtin_ctz((unsigned int)m);
  mint w = mint(fps::ntt_root()).pow((mint::get_mod() - 1) >> (log + 1));
  mint wi = w.inv();
  vector<int> rev(m);
  for (int i = 0; i < rev.size(); i++) rev[i] = (rev[i / 2] / 2) | ((i & 1) << (log - 1));
  mint inv2 = mint(2).inv();

  f.resize(m);
  f.ntt();
  auto rec = [&](auto& rec, long long n) -> void {
    if (n < m) {
      f.intt();
      f = f.inv(n + m);
      f >>= n;
      f.ntt();
      return;
    }
    f.ntt_doubling();
    assert(f.size() == 2 * m);
    auto f1 = f;
    for (int i = 0; i < m; i++) swap(f1[i << 1], f1[(i << 1) | 1]);
    for (int i = 0; i < m; i++) f[i] = f[i << 1] * f[(i << 1) | 1];
    f.resize(m);
    rec(rec, (n - m + 1) / 2);
    if (((n - m) & 1) == 0) {
      f.resize(2 * m);
      for (int i = m - 1; i >= 0; i--) {
        f[(i << 1) | 1] = f[i];
        f[i << 1] = f[i];
      }
    } else {
      mint p = 1;
      for (auto i : rev) f[i] *= p, p *= w;
      f.resize(2 * m);
      for (int i = m - 1; i >= 0; i--) {
        f[(i << 1) | 1] = -f[i];
        f[i << 1] = f[i];
      }
    }
    for (int i = 0; i < 2 * m; i++) f[i] *= f1[i];
    auto odd = fps(f.begin() + m, f.end());
    odd.intt();
    mint p = 1;
    for (int i = 0; i < m; i++) odd[i] *= p, p *= wi;
    odd.ntt();
    f.resize(m);
    for (int i = 0; i < m; i++) f[i] = (f[i] - odd[i]) * inv2;
  };
  rec(rec, n);
  f.intt();
  f.resize(k);
  return f;
}
/**
 * @brief Inverse の次数シフト
 * @docs docs/fps/inverse-shift.md
 */
#line 6 "fps/linearly-recurrent-sequence.hpp"

// a[i]=sum[j=1~d]c[j]a[i-j], i>=d
// find a[n]
template <class mint>
mint LinearyRecurrentSequence(FormalPowerSeries<mint> a, FormalPowerSeries<mint> c, long long n) {
  assert(a.size() == c.size());
  if (n < a.size()) return a[n];
  while (!c.empty() && c.back() == 0) c.pop_back();
  if (c.size() < a.size()) {
    int z = a.size() - c.size();
    n -= z;
    a.erase(a.begin(), a.begin() + z);
  }
  int d = c.size();
  FormalPowerSeries<mint> q(d + 1);
  q[0] = 1;
  for (int i = 0; i < d; i++) q[i + 1] = -c[i];
  auto p = (a * q).pre(d);
  return BostanMori(p, q, n);
}

// a[i]=sum[j=1~d]c[j]a[i-j], i>=d
// find a[n],a[n+1],...,a[n+k-1]
template <class mint>
FormalPowerSeries<mint> LinearyRecurrentSequence(FormalPowerSeries<mint> a, FormalPowerSeries<mint> c, long long n, int k) {
  assert(a.size() == c.size());
  if (n + k < a.size()) {
    a >>= (int)n;
    return a.pre(k);
  }
  while (!c.empty() && c.back() == 0) c.pop_back();
  int d = c.size();
  FormalPowerSeries<mint> ret{};
  if (c.size() < a.size()) {
    int z = a.size() - c.size();
    if (n < z) {
      ret.reserve(k);
      for (int i = n; i < z; i++) ret.push_back(a[i]);
    }
    n -= z;
    if (n < 0) {
      k += n;
      n = 0;
    }
    a >>= z;
  }
  FormalPowerSeries<mint> q(d + 1);
  q[0] = 1;
  for (int i = 0; i < d; i++) q[i + 1] = -c[i];
  auto p = (a * q).pre(d);
  if (n < a.size()) {
    p *= q.inv(n + k);
    p >>= n;
  } else {
    p *= (InverseShift(q, n) * q).pre(d);
    p %= q;
    p *= q.inv(k);
  }
  p.resize(k);
  if (ret.empty()) {
    return p;
  } else {
    for (auto v : p) ret.push_back(v);
    return ret;
  }
}

template <class mint>
mint LinearyRecurrentSequence(FormalPowerSeries<mint> a, long long n) {
  if (n < a.size()) return a[n];
  auto b = BerlekampMassey(a);
  int d = b.size() - 1;
  a.resize(d);
  int z = 0;
  while (b.back() == 0) b.pop_back(), z++;
  a >>= z;
  n -= z;
  d -= z;
  return BostanMori((a * b).pre(b.size()), b, n);
}

/**
 * @brief 線形漸化式用
 * @docs docs/fps/linearly-recurrent-sequence.md
 */

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