If you want a simple pseudo random sequence of N that just looks random (pseudocode):

  int start = getRandomInt(seed) % N;
  double k = N / 1.618;
  k = k % 2 == 0 ? k - 1 : k; // k and N must be coprime
  int nextElement = k * start % N;
  start++;
  

This will jump wildly across your sequence and visit all elements (as k and N are chosen coprime). No need to store a full array. Is it statistically random? Of course not, e.g. you will never see two values close to each other right after another.