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%)


link printf, factors
 
$define MAX_N 20
$define TIMES 1000000
$define RAND_MAX 2147483647
 
procedure expected(n)
    local sum := 0
    every i := 1 to n do
        sum +:= factorial(n) / (n ^ i) / factorial(n - i)
    return sum
end
 
procedure test(n, times)
    local i, count := 0, x, bits
    every i := 0 to times-1 do {
	x := 1
	bits := 0
	while iand(bits, x)=0 do {
            count +:= 1
            bits := ior(bits, x)
            x := ishift(1 , ?n-1)
        }
    }
    return count
end
 
procedure main(void)
    local n, cnt, avg, theory, diff
    write(" n\tavg\texp.\tdiff\n", repl("-",29))
    every n := 1 to MAX_N do {
        cnt := test(n, TIMES)
        avg := real(cnt) / TIMES
        theory := expected(n)
        diff := (avg / theory - 1) * 100
        printf("%2d %8.4r %8.4r %6.3r%%\n", n, avg, theory, diff)
    }
    return 0
end