Let f be a uniformly-randomly chosen mapping from the numbers 1..N to the numbers 1..N (note: not necessarily a permutation of 1..N; the mapping could produce a number in more than one way or not at all). At some point, the sequence 1, f(1), f(f(1))... will contain a repetition, a number that occurring for the second time in the sequence.



Task

Write a program or a script that estimates, for each N, the average length until the first such repetition.

Also calculate this expected length using an analytical formula, and optionally compare the simulated result with the theoretical one.



This problem comes from the end of Donald Knuth's Christmas tree lecture 2011.

Example of expected output:

 N    average    analytical    (error)
===  =========  ============  =========
  1     1.0000        1.0000  (  0.00%)
  2     1.4992        1.5000  (  0.05%)
  3     1.8784        1.8889  (  0.56%)
  4     2.2316        2.2188  (  0.58%)
  5     2.4982        2.5104  (  0.49%)
  6     2.7897        2.7747  (  0.54%)
  7     3.0153        3.0181  (  0.09%)
  8     3.2429        3.2450  (  0.07%)
  9     3.4536        3.4583  (  0.14%)
 10     3.6649        3.6602  (  0.13%)
 11     3.8091        3.8524  (  1.12%)
 12     3.9986        4.0361  (  0.93%)
 13     4.2074        4.2123  (  0.12%)
 14     4.3711        4.3820  (  0.25%)
 15     4.5275        4.5458  (  0.40%)
 16     4.6755        4.7043  (  0.61%)
 17     4.8877        4.8579  (  0.61%)
 18     4.9951        5.0071  (  0.24%)
 19     5.1312        5.1522  (  0.41%)
 20     5.2699        5.2936  (  0.45%)


import random, math, strformat
randomize()
 
const
  maxN = 20
  times = 1_000_000
 
proc factorial(n: int): float =
  result = 1
  for i in 1 .. n:
    result *= i.float
 
proc expected(n: int): float =
  for i in 1 .. n:
    result += factorial(n) / pow(n.float, i.float) / factorial(n - i)
 
proc test(n, times: int): int =
  for i in 1 .. times:
    var
      x = 1
      bits = 0
    while (bits and x) == 0:
      inc result
      bits = bits or x
      x = 1 shl rand(n - 1)
 
echo " n\tavg\texp.\tdiff"
echo "-------------------------------"
for n in 1 .. maxN:
  let cnt = test(n, times)
  let avg = cnt.float / times
  let theory = expected(n)
  let diff = (avg / theory - 1) * 100
  echo fmt"{n:2} {avg:8.4f} {theory:8.4f} {diff:6.3f}%"