BM板子
BM线性递推
玄学玩意
struct LinearRecurrence
{
using int64 = long long;
using vec = std::vector<int64>; static void extand(vec& a, size_t d, int64 value = )
{
if (d <= a.size()) return;
a.resize(d, value);
}
static vec BerlekampMassey(const vec& s, int64 mod)
{
std::function<int64(int64)> inverse = [&](int64 a) {
return a == ? : (int64)(mod - mod / a) * inverse(mod % a) % mod;
};
vec A = {}, B = {};
int64 b = s[];
for (size_t i = , m = ; i < s.size(); ++i, m++)
{
int64 d = ;
for (size_t j = ; j < A.size(); ++j)
{
d += A[j] * s[i - j] % mod;
}
if (!(d %= mod)) continue;
if ( * (A.size() - ) <= i)
{
auto temp = A;
extand(A, B.size() + m);
int64 coef = d * inverse(b) % mod;
for (size_t j = ; j < B.size(); ++j)
{
A[j + m] -= coef * B[j] % mod;
if (A[j + m] < ) A[j + m] += mod;
}
B = temp, b = d, m = ;
}
else
{
extand(A, B.size() + m);
int64 coef = d * inverse(b) % mod;
for (size_t j = ; j < B.size(); ++j)
{
A[j + m] -= coef * B[j] % mod;
if (A[j + m] < ) A[j + m] += mod;
}
}
}
return A;
}
static void exgcd(int64 a, int64 b, int64& g, int64& x, int64& y)
{
if (!b)
x = , y = , g = a;
else
{
exgcd(b, a % b, g, y, x);
y -= x * (a / b);
}
}
static int64 crt(const vec& c, const vec& m)
{
int n = c.size();
int64 M = , ans = ;
for (int i = ; i < n; ++i) M *= m[i];
for (int i = ; i < n; ++i)
{
int64 x, y, g, tm = M / m[i];
exgcd(tm, m[i], g, x, y);
ans = (ans + tm * x * c[i] % M) % M;
}
return (ans + M) % M;
}
static vec ReedsSloane(const vec& s, int64 mod)
{
auto inverse = [](int64 a, int64 m) {
int64 d, x, y;
exgcd(a, m, d, x, y);
return d == ? (x % m + m) % m : -;
};
auto L = [](const vec& a, const vec& b) {
int da = (a.size() > || (a.size() == && a[])) ? a.size() - : -;
int db = (b.size() > || (b.size() == && b[])) ? b.size() - : -;
return std::max(da, db + );
};
auto prime_power = [&](const vec& s, int64 mod, int64 p, int64 e) {
// linear feedback shift register mod p^e, p is prime
std::vector<vec> a(e), b(e), an(e), bn(e), ao(e), bo(e);
vec t(e), u(e), r(e), to(e, ), uo(e), pw(e + );
;
pw[] = ;
for (int i = pw[] = ; i <= e; ++i) pw[i] = pw[i - ] * p;
for (int64 i = ; i < e; ++i)
{
a[i] = {pw[i]}, an[i] = {pw[i]};
b[i] = {}, bn[i] = {s[] * pw[i] % mod};
t[i] = s[] * pw[i] % mod;
if (t[i] == )
{
t[i] = , u[i] = e;
}
else
{
for (u[i] = ; t[i] % p == ; t[i] /= p, ++u[i])
;
}
}
for (size_t k = ; k < s.size(); ++k)
{
for (int g = ; g < e; ++g)
{
if (L(an[g], bn[g]) > L(a[g], b[g]))
{
ao[g] = a[e - - u[g]];
bo[g] = b[e - - u[g]];
to[g] = t[e - - u[g]];
uo[g] = u[e - - u[g]];
r[g] = k - ;
}
}
a = an, b = bn;
for (int o = ; o < e; ++o)
{
int64 d = ;
for (size_t i = ; i < a[o].size() && i <= k; ++i)
{
d = (d + a[o][i] * s[k - i]) % mod;
}
if (d == )
{
t[o] = , u[o] = e;
}
else
{
for (u[o] = , t[o] = d; t[o] % p == ; t[o] /= p, ++u[o])
;
int g = e - - u[o];
if (L(a[g], b[g]) == )
{
extand(bn[o], k + );
bn[o][k] = (bn[o][k] + d) % mod;
}
else
{
int64 coef = t[o] * inverse(to[g], mod) % mod * pw[u[o] - uo[g]] % mod;
int m = k - r[g];
extand(an[o], ao[g].size() + m);
extand(bn[o], bo[g].size() + m);
for (size_t i = ; i < ao[g].size(); ++i)
{
an[o][i + m] -= coef * ao[g][i] % mod;
if (an[o][i + m] < ) an[o][i + m] += mod;
}
while (an[o].size() && an[o].back() == ) an[o].pop_back();
for (size_t i = ; i < bo[g].size(); ++i)
{
bn[o][i + m] -= coef * bo[g][i] % mod;
if (bn[o][i + m] < ) bn[o][i + m] -= mod;
}
while (bn[o].size() && bn[o].back() == ) bn[o].pop_back();
}
}
}
}
return std::make_pair(an[], bn[]);
}; std::vector<std::tuple<int64, int64, int>> fac;
for (int64 i = ; i * i <= mod; ++i)
{
if (mod % i == )
{
int64 cnt = , pw = ;
while (mod % i == ) mod /= i, ++cnt, pw *= i;
fac.emplace_back(pw, i, cnt);
}
}
if (mod > ) fac.emplace_back(mod, mod, );
std::vector<vec> as;
size_t n = ;
for (auto&& x : fac)
{
int64 mod, p, e;
vec a, b;
std::tie(mod, p, e) = x;
auto ss = s;
for (auto&& x : ss) x %= mod;
std::tie(a, b) = prime_power(ss, mod, p, e);
as.emplace_back(a);
n = std::max(n, a.size());
}
vec a(n), c(as.size()), m(as.size());
for (size_t i = ; i < n; ++i)
{
for (size_t j = ; j < as.size(); ++j)
{
m[j] = std::get<>(fac[j]);
c[j] = i < as[j].size() ? as[j][i] : ;
}
a[i] = crt(c, m);
}
return a;
} LinearRecurrence(const vec& s, const vec& c, int64 mod) : init(s), trans(c), mod(mod), m(s.size()) {}
LinearRecurrence(const vec& s, int64 mod, bool is_prime = true) : mod(mod)
{
vec A;
if (is_prime)
A = BerlekampMassey(s, mod);
else
A = ReedsSloane(s, mod);
if (A.empty()) A = {};
m = A.size() - ;
trans.resize(m);
for (int i = ; i < m; ++i)
{
trans[i] = (mod - A[i + ]) % mod;
}
std::reverse(trans.begin(), trans.end());
init = {s.begin(), s.begin() + m};
}
int64 calc(int64 n)
{
if (mod == ) return ;
if (n < m) return init[n];
vec v(m), u(m << );
int msk = !!n;
for (int64 m = n; m > ; m >>= ) msk <<= ;
v[] = % mod;
for (int x = ; msk; msk >>= , x <<= )
{
std::fill_n(u.begin(), m * , );
x |= !!(n & msk);
if (x < m)
u[x] = % mod;
else
{ // can be optimized by fft/ntt
for (int i = ; i < m; ++i)
{
for (int j = , t = i + (x & ); j < m; ++j, ++t)
{
u[t] = (u[t] + v[i] * v[j]) % mod;
}
}
for (int i = m * - ; i >= m; --i)
{
for (int j = , t = i - m; j < m; ++j, ++t)
{
u[t] = (u[t] + trans[j] * u[i]) % mod;
}
}
}
v = {u.begin(), u.begin() + m};
}
int64 ret = ;
for (int i = ; i < m; ++i)
{
ret = (ret + v[i] * init[i]) % mod;
}
return ret;
} vec init, trans;
int64 mod;
int m;
};
接口： LinearRecurrence solver(f, mod, false);//f前若干项vector,mod模数，最后一个是否为质数
solver.calc(n)；//计算第n项
#include<bits/stdc++.h> struct LinearRecurrence
{
using int64 = long long;
using vec = std::vector<int64>; static void extand(vec& a, size_t d, int64 value = )
{
if (d <= a.size()) return;
a.resize(d, value);
}
static vec BerlekampMassey(const vec& s, int64 mod)
{
std::function<int64(int64)> inverse = [&](int64 a) {
return a == ? : (int64)(mod - mod / a) * inverse(mod % a) % mod;
};
vec A = {}, B = {};
int64 b = s[];
for (size_t i = , m = ; i < s.size(); ++i, m++)
{
int64 d = ;
for (size_t j = ; j < A.size(); ++j)
{
d += A[j] * s[i - j] % mod;
}
if (!(d %= mod)) continue;
if ( * (A.size() - ) <= i)
{
auto temp = A;
extand(A, B.size() + m);
int64 coef = d * inverse(b) % mod;
for (size_t j = ; j < B.size(); ++j)
{
A[j + m] -= coef * B[j] % mod;
if (A[j + m] < ) A[j + m] += mod;
}
B = temp, b = d, m = ;
}
else
{
extand(A, B.size() + m);
int64 coef = d * inverse(b) % mod;
for (size_t j = ; j < B.size(); ++j)
{
A[j + m] -= coef * B[j] % mod;
if (A[j + m] < ) A[j + m] += mod;
}
}
}
return A;
}
static void exgcd(int64 a, int64 b, int64& g, int64& x, int64& y)
{
if (!b)
x = , y = , g = a;
else
{
exgcd(b, a % b, g, y, x);
y -= x * (a / b);
}
}
static int64 crt(const vec& c, const vec& m)
{
int n = c.size();
int64 M = , ans = ;
for (int i = ; i < n; ++i) M *= m[i];
for (int i = ; i < n; ++i)
{
int64 x, y, g, tm = M / m[i];
exgcd(tm, m[i], g, x, y);
ans = (ans + tm * x * c[i] % M) % M;
}
return (ans + M) % M;
}
static vec ReedsSloane(const vec& s, int64 mod)
{
auto inverse = [](int64 a, int64 m) {
int64 d, x, y;
exgcd(a, m, d, x, y);
return d == ? (x % m + m) % m : -;
};
auto L = [](const vec& a, const vec& b) {
int da = (a.size() > || (a.size() == && a[])) ? a.size() - : -;
int db = (b.size() > || (b.size() == && b[])) ? b.size() - : -;
return std::max(da, db + );
};
auto prime_power = [&](const vec& s, int64 mod, int64 p, int64 e) {
// linear feedback shift register mod p^e, p is prime
std::vector<vec> a(e), b(e), an(e), bn(e), ao(e), bo(e);
vec t(e), u(e), r(e), to(e, ), uo(e), pw(e + );
;
pw[] = ;
for (int i = pw[] = ; i <= e; ++i) pw[i] = pw[i - ] * p;
for (int64 i = ; i < e; ++i)
{
a[i] = {pw[i]}, an[i] = {pw[i]};
b[i] = {}, bn[i] = {s[] * pw[i] % mod};
t[i] = s[] * pw[i] % mod;
if (t[i] == )
{
t[i] = , u[i] = e;
}
else
{
for (u[i] = ; t[i] % p == ; t[i] /= p, ++u[i])
;
}
}
for (size_t k = ; k < s.size(); ++k)
{
for (int g = ; g < e; ++g)
{
if (L(an[g], bn[g]) > L(a[g], b[g]))
{
ao[g] = a[e - - u[g]];
bo[g] = b[e - - u[g]];
to[g] = t[e - - u[g]];
uo[g] = u[e - - u[g]];
r[g] = k - ;
}
}
a = an, b = bn;
for (int o = ; o < e; ++o)
{
int64 d = ;
for (size_t i = ; i < a[o].size() && i <= k; ++i)
{
d = (d + a[o][i] * s[k - i]) % mod;
}
if (d == )
{
t[o] = , u[o] = e;
}
else
{
for (u[o] = , t[o] = d; t[o] % p == ; t[o] /= p, ++u[o])
;
int g = e - - u[o];
if (L(a[g], b[g]) == )
{
extand(bn[o], k + );
bn[o][k] = (bn[o][k] + d) % mod;
}
else
{
int64 coef = t[o] * inverse(to[g], mod) % mod * pw[u[o] - uo[g]] % mod;
int m = k - r[g];
extand(an[o], ao[g].size() + m);
extand(bn[o], bo[g].size() + m);
for (size_t i = ; i < ao[g].size(); ++i)
{
an[o][i + m] -= coef * ao[g][i] % mod;
if (an[o][i + m] < ) an[o][i + m] += mod;
}
while (an[o].size() && an[o].back() == ) an[o].pop_back();
for (size_t i = ; i < bo[g].size(); ++i)
{
bn[o][i + m] -= coef * bo[g][i] % mod;
if (bn[o][i + m] < ) bn[o][i + m] -= mod;
}
while (bn[o].size() && bn[o].back() == ) bn[o].pop_back();
}
}
}
}
return std::make_pair(an[], bn[]);
}; std::vector<std::tuple<int64, int64, int>> fac;
for (int64 i = ; i * i <= mod; ++i)
{
if (mod % i == )
{
int64 cnt = , pw = ;
while (mod % i == ) mod /= i, ++cnt, pw *= i;
fac.emplace_back(pw, i, cnt);
}
}
if (mod > ) fac.emplace_back(mod, mod, );
std::vector<vec> as;
size_t n = ;
for (auto&& x : fac)
{
int64 mod, p, e;
vec a, b;
std::tie(mod, p, e) = x;
auto ss = s;
for (auto&& x : ss) x %= mod;
std::tie(a, b) = prime_power(ss, mod, p, e);
as.emplace_back(a);
n = std::max(n, a.size());
}
vec a(n), c(as.size()), m(as.size());
for (size_t i = ; i < n; ++i)
{
for (size_t j = ; j < as.size(); ++j)
{
m[j] = std::get<>(fac[j]);
c[j] = i < as[j].size() ? as[j][i] : ;
}
a[i] = crt(c, m);
}
return a;
} LinearRecurrence(const vec& s, const vec& c, int64 mod) : init(s), trans(c), mod(mod), m(s.size()) {}
LinearRecurrence(const vec& s, int64 mod, bool is_prime = true) : mod(mod)
{
vec A;
if (is_prime)
A = BerlekampMassey(s, mod);
else
A = ReedsSloane(s, mod);
if (A.empty()) A = {};
m = A.size() - ;
trans.resize(m);
for (int i = ; i < m; ++i)
{
trans[i] = (mod - A[i + ]) % mod;
}
std::reverse(trans.begin(), trans.end());
init = {s.begin(), s.begin() + m};
}
int64 calc(int64 n)
{
if (mod == ) return ;
if (n < m) return init[n];
vec v(m), u(m << );
int msk = !!n;
for (int64 m = n; m > ; m >>= ) msk <<= ;
v[] = % mod;
for (int x = ; msk; msk >>= , x <<= )
{
std::fill_n(u.begin(), m * , );
x |= !!(n & msk);
if (x < m)
u[x] = % mod;
else
{ // can be optimized by fft/ntt
for (int i = ; i < m; ++i)
{
for (int j = , t = i + (x & ); j < m; ++j, ++t)
{
u[t] = (u[t] + v[i] * v[j]) % mod;
}
}
for (int i = m * - ; i >= m; --i)
{
for (int j = , t = i - m; j < m; ++j, ++t)
{
u[t] = (u[t] + trans[j] * u[i]) % mod;
}
}
}
v = {u.begin(), u.begin() + m};
}
int64 ret = ;
for (int i = ; i < m; ++i)
{
ret = (ret + v[i] * init[i]) % mod;
}
return ret;
} vec init, trans;
int64 mod;
int m;
};
using namespace std;
typedef long long ll;
ll A[];
ll mod=1e9;
void init()
{
A[]=A[]=;
A[]=;
for(int i=;i<=;i++){
A[i]=(A[i-]+A[i-])%mod;
}
}
ll qp(ll x,ll n)
{
ll ans=;
while(n){
if(n&)ans=(ans*x)%mod;
x=(x*x)%mod;
n>>=;
}
return ans;
}
int main()
{
init();
ll n,m;
scanf("%lld%lld",&n,&m);
vector<ll> v;
ll t=;
for(int i=;i<=min(n,50ll);i++){
t=(t+qp(A[i],m))%mod;
v.push_back(t);
}
LinearRecurrence L(v, mod, false);
cout<<L.calc(n)<<'\n'; }
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