f(f(x))=-x.cpp

#include <iostream>
#include <algorithm>

int f[100];
int mx[100]; // mx[x] = (-x modulo N)

int main() {
    for (int i = 0; i < N; ++i) {
        f[i] = i;
        mx[i] = (N - i) % N;
    }

    do {

        bool found = true;
        for (int i = 0; i < N; ++i) {
            if (f[f[i]] != mx[i]) {
                found = false;
                break;
            }
        }
        if (found) {
            std::cout << N << ": ";
            for (int i = 0; i < N; ++i) {
                std::cout << f[i] << ';';
            }
            std::cout << std::endl;
            exit(0);
        }
    } while(N <= 1 || std::next_permutation(f+1, f+N));

    std::cout << N << ": not found." << std::endl;
}

output.sh

for i in {1..16}; do g++ -DN="$i" -O3 -DNDEBUG x.cpp && ./a.out  ; done

1: 0;
2: 0;1;
3: not found.
4: not found.
5: 0;2;4;1;3;
6: 0;2;5;3;1;4;
7: not found.
8: not found.
9: 0;2;8;4;6;3;5;1;7;
10: 0;2;9;4;7;5;3;6;1;8;
11: not found.
12: not found.
13: 0;2;12;4;10;6;8;5;7;3;9;1;11;
14: 0;2;13;4;11;6;9;7;5;8;3;10;1;12;
15: not found.