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%)


Const MAX = 20
Const ITER = 1000000
 
Function expected(n As Long) As Double
    Dim sum As Double
    For i = 1 To n
        sum = sum + WorksheetFunction.Fact(n) / n ^ i / WorksheetFunction.Fact(n - i)
    Next i
    expected = sum
End Function
 
Function test(n As Long) As Double
    Dim count As Long
    Dim x As Long, bits As Long
    For i = 1 To ITER
        x = 1
        bits = 0
        Do While Not bits And x
            count = count + 1
            bits = bits Or x
            x = 2 ^ (Int(n * Rnd()))
        Loop
    Next i
    test = count / ITER
End Function
 
Public Sub main()
    Dim n As Long
    Debug.Print " n     avg.     exp.  (error%)"
    Debug.Print "==   ======   ======  ========"
    For n = 1 To MAX
        av = test(n)
        ex = expected(n)
        Debug.Print Format(n, "@@"); "  "; Format(av, "0.0000"); "   ";
        Debug.Print Format(ex, "0.0000"); "  ("; Format(Abs(1 - av / ex), "0.000%"); ")"
    Next n
End Sub