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%)


      @% = &2040A
      MAX_N = 20
      TIMES = 1000000
 
      FOR n = 1 TO MAX_N
        avg = FNtest(n, TIMES)
        theory = FNanalytical(n)
        diff = (avg / theory - 1) * 100
        PRINT STR$(n), avg, theory, diff "%"
      NEXT
      END
 
      DEF FNanalytical(n)
      LOCAL i, s
      FOR i = 1 TO n
        s += FNfactorial(n) / n^i / FNfactorial(n-i)
      NEXT
      = s
 
      DEF FNtest(n, times)
      LOCAL i, b, c, x
      FOR i = 1 TO times
        x = 1 : b = 0
        WHILE (b AND x) = 0
          c += 1
          b OR= x
          x = 1 << (RND(n) - 1)
        ENDWHILE
      NEXT
      = c / times
 
      DEF FNfactorial(n)
      IF n=1 OR n=0 THEN =1 ELSE = n * FNfactorial(n-1)