#include #include #include #include #include #include #include #include #include #include #include #include #include #include #ifndef templete #define rep(i,a,b) for(int i=a;i=b;i--) #define fore(i,a) for(auto &i:a) #define all(x) (x).begin(),(x).end() //#include //using namespace boost::multiprecision; using namespace std; using namespace atcoder; //using atmint = modint998244353; using atmint = modint; using Graph = vector>; using P = pair; //#pragma GCC optimize ("-O3") using namespace std; void _main(); int main() { cin.tie(0); ios::sync_with_stdio(false); _main(); } typedef long long ll; const int inf = INT_MAX / 2; const ll infl = 1LL << 60; templatebool chmax(T& a, const T& b) { if (a < b) { a = b; return 1; } return 0; } templatebool chmin(T& a, const T& b) { if (b < a) { a = b; return 1; } return 0; } //--------------------------------------------------------------------------------------------------- template struct ModInt { static const int Mod = MOD; unsigned x; ModInt() : x(0) { } ModInt(signed sig) { x = sig < 0 ? sig % MOD + MOD : sig % MOD; } ModInt(signed long long sig) { x = sig < 0 ? sig % MOD + MOD : sig % MOD; } int get() const { return (int)x; } ModInt &operator+=(ModInt that) { if ((x += that.x) >= MOD) x -= MOD; return *this; } ModInt &operator-=(ModInt that) { if ((x += MOD - that.x) >= MOD) x -= MOD; return *this; } ModInt &operator*=(ModInt that) { x = (unsigned long long)x * that.x % MOD; return *this; } ModInt &operator/=(ModInt that) { return *this *= that.inverse(); } ModInt operator+(ModInt that) const { return ModInt(*this) += that; } ModInt operator-(ModInt that) const { return ModInt(*this) -= that; } ModInt operator*(ModInt that) const { return ModInt(*this) *= that; } ModInt operator/(ModInt that) const { return ModInt(*this) /= that; } ModInt inverse() const { long long a = x, b = MOD, u = 1, v = 0; while (b) { long long t = a / b; a -= t * b; std::swap(a, b); u -= t * v; std::swap(u, v); } return ModInt(u); } bool operator==(ModInt that) const { return x == that.x; } bool operator!=(ModInt that) const { return x != that.x; } ModInt operator-() const { ModInt t; t.x = x == 0 ? 0 : Mod - x; return t; } }; template ostream& operator<<(ostream& st, const ModInt a) { st << a.get(); return st; }; template ModInt operator^(ModInt a, unsigned long long k) { ModInt r = 1; while (k) { if (k & 1) r *= a; a *= a; k >>= 1; } return r; } template struct Comb { vector fac, ifac; Comb(){fac.resize(FAC_MAX,1);ifac.resize(FAC_MAX,1);rep(i,1,FAC_MAX)fac[i]=fac[i-1]*i; ifac[FAC_MAX-1]=T(1)/fac[FAC_MAX-1];rrep(i,FAC_MAX-2,1)ifac[i]=ifac[i+1]*T(i+1);} T aPb(int a, int b) { if (b < 0 || a < b) return T(0); return fac[a] * ifac[a - b]; } T aCb(int a, int b) { if (b < 0 || a < b) return T(0); return fac[a] * ifac[a - b] * ifac[b]; } T nHk(int n, int k) { if (n == 0 && k == 0) return T(1); if (n <= 0 || k < 0) return 0; return aCb(n + k - 1, k); } // nHk = (n+k-1)Ck : n is separator T pairCombination(int n) {if(n%2==1)return T(0);return fac[n]*ifac[n/2]/(T(2)^(n/2));} // combination of paris for n com.aCb(h+w-2,h-1); }; //typedef ModInt<1000000007> mint; typedef ModInt<998244353> mint; //typedef ModInt<1000000000> mint; Comb com; //vector dp(n+1,vector(n+1,vector(n+1,0))); //vector dp(n+1,vector(n+1,0)); std::random_device seed_gen; std::mt19937 engine(seed_gen()); string ye = "Yes"; string no = "No"; string draw = "Draw"; #endif // templete //--------------------------------------------------------------------------------------------------- void _main() { ll k; cin >> k; ll maxtime = 100000; vector dp(k+1,vector(maxtime,0)); vector seen(k+1,vector(maxtime,0)); priority_queue,greater

>q; q.push({0,0}); seen[0][0] = 1; dp[0][0] = 1.0; while(q.size()){ ll time = q.top().first; ll tot = q.top().second; q.pop(); rep(i,1,7){ ll ntime = time + 1; ll ntot = tot + i; if(ntime >= maxtime)continue; if(ntot > k)ntot = 0; dp[ntot][ntime] += dp[tot][time] / 6.0; if(seen[ntot][ntime] == 0 && ntot != k){ seen[ntot][ntime] = 1; q.push({ntime,ntot}); } } } /* rep(i,0,k+1){ rep(j,0,maxtime)cout << dp[i][j] << " "; cout << endl; } */ double ans = 0; rep(i,0,maxtime)ans += dp[k][i] * i; cout << fixed << setprecision(20); cout << ans << endl; }