#include //#include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include using namespace std; //using namespace atcoder; using ll = long long; using ld = long double; using pii = pair; using pll = pair; using vll = vector; using vvll = vector>; using vvvll = vector>>; using vii = vector; using vvii = vector>; using vvvii = vector>>; using vdd = vector; using vvdd = vector>; using vvvdd = vector>>; using vbb = vector; using vvbb = vector>; using vvvbb = vector>>; #define pb push_back #define mp make_pair #define sc second #define fr first #define endl '\n' #define stpr std::fixed << setprecision #define cYES cout << "YES" << endl #define cNO cout << "NO" << endl #define cYes cout << "Yes" << endl #define cNo cout << "No" << endl #define rep(i, n) for (ll i = 0; i < (n); ++i) #define drep(i, a, b, d) for (ll i = (a); i <= (b); i += d) #define Rep(i, a, b) for (ll i = (a); i < (b); ++i) #define rrep(i, n) for (ll i = n - 1; i >= 0; i--) #define drrep(i, a, b, d) for (ll i = (a); i >= (b); i -= d) #define rRep(i, a, b) for (ll i = a; i >= b; i--) #define crep(i) for (char i = 'a'; i <= 'z'; ++i) #define Crep(i) for (char i = 'A'; i <= 'Z'; ++i) #define ALL(x) (x).begin(), (x).end() #define rALL(x) (x).rbegin(), (x).rend() #define sort2(A, N) \ sort(A, A + N, \ [](const pii &a, const pii &b) { return a.second < b.second; }); #define debug(v) \ cout << #v << ":"; \ for (auto x : v) { \ cout << x << ' '; \ } \ cout << endl; int ctoi(const char c) { if ('0' <= c && c <= '9') return (c - '0'); return -1; } ll gcd(ll a, ll b) { return (b == 0 ? a : gcd(b, a % b)); } ll lcm(ll a, ll b) { return a * b / gcd(a, b); } constexpr ll MOD = 1000000007; constexpr ll INF = 1000000011; constexpr ll MOD2 = 998244353; constexpr ll LINF = 1001002003004005006ll; constexpr ld EPS = 10e-10; template inline bool chmax(T &lhs, const U &rhs) { if (lhs < rhs) { lhs = rhs; return 1; } return 0; } template inline bool chmin(T &lhs, const U &rhs) { if (lhs > rhs) { lhs = rhs; return 1; } return 0; } template istream &operator>>(istream &is, vector &v) { for (auto &&x : v) is >> x; return is; } template istream &operator>>(istream &is, pair &p) { is >> p.first; is >> p.second; return is; } template ostream &operator<<(ostream &os, const pair &p) { os << p.first << ' ' << p.second; return os; } template ostream &operator<<(ostream &os, vector &v) { for (auto i = begin(v); i != end(v); ++i) { if (i != begin(v)) os << ' '; os << *i; } return os; } // 連立方程式の形をそのまま入れる vdd gauss_jordan(vvdd &A, const vdd &b) { int n_ = A.size(), m_ = A[0].size(); if (n_ < m_) { return vdd(); } vvdd B(n_, vdd(m_ + 1)); auto c = b; for (int i = 0; i < n_; i++) { for (int j = 0; j < m_; j++) { B[i][j] = A[i][j]; } } for (int i = 0; i < n_; i++) { B[i][m_] = b[i]; } for (int i = 0; i < m_; i++) { int pivot = i; for (int j = i; j < n_; j++) { if (abs(B[j][i]) > abs(B[pivot][i])) { pivot = j; } } swap(B[i], B[pivot]); swap(c[i], c[pivot]); if (abs(B[i][i]) < EPS) { return vdd(); } for (int j = i + 1; j <= m_; j++) { B[i][j] /= B[i][i]; } for (int j = 0; j < n_; j++) { if (i != j) { for (int k = i + 1; k <= m_; k++) { B[j][k] -= B[j][i] * B[i][k]; } } } } vdd x(m_); for (int i = 0; i < m_; i++) { x[i] = B[i][m_]; } bool T = 0; for (int i = m_; i < n_; i++) { ld sum = 0; for (int j = 0; j < m_; j++) { sum += x[j] * B[i][j]; } if (sum - c[i] > EPS) { return vdd(); } } return x; } int main() { ll K; cin >> K; vvdd A(K + 1, vdd(K + 1, 0)); vdd B(K + 1, -1.0); rep(i, K) { A[i][i] = -1.0; rep(j, 6) { ll t = i + j + 1; if (t > K) { t = 0; } A[i][t] += 1.0 / 6.0; } } A[K][K] = 1; B[K] = 0; auto V = gauss_jordan(A, B); cout << stpr(15) << V[0] << endl; }