# Average loop length

Average loop length
You are encouraged to solve this task according to the task description, using any language you may know.

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.

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%)

procedure Avglen is
package IIO is new Ada.Text_IO.Integer_IO (Positive); use IIO;
package LFIO is new Ada.Text_IO.Float_IO (Long_Float); use LFIO;
subtype FactN is Natural range 0..20;
TESTS : constant Natural := 1_000_000;

function Factorial (N : FactN) return Long_Float is
Result : Long_Float := 1.0;
begin
for I in 2..N loop Result := Result * Long_Float(I); end loop;
return Result;
end Factorial;

function Analytical (N : FactN) return Long_Float is
Sum : Long_Float := 0.0;
begin
for I in 1..N loop
Sum := Sum + Factorial(N) / Factorial(N - I) / Long_Float(N)**I;
end loop;
return Sum;
end Analytical;

function Experimental (N : FactN) return Long_Float is
subtype RandInt is Natural range 1..N;
package Random is new Ada.Numerics.Discrete_Random(RandInt);
seed : Random.Generator;
Num : RandInt;
count : Natural := 0;
bits : array(RandInt'Range) of Boolean;
begin
Random.Reset(seed);
for run in 1..TESTS loop
bits := (others => false);
for I in RandInt'Range loop
Num := Random.Random(seed); exit when bits(Num);
bits(Num) := True; count := count + 1;
end loop;
end loop;
return Long_Float(count)/Long_Float(TESTS);
end Experimental;

A, E, err : Long_Float;
begin
Put_Line(" N avg calc  %diff");
for I in 1..20 loop
A := Analytical(I); E := Experimental(I); err := abs(E-A)/A*100.0;
Put(I, Width=>2); Put(E ,Aft=>4, exp=>0); Put(A, Aft=>4, exp=>0);
Put(err, Fore=>3, Aft=>3, exp=>0); New_line;
end loop;
end Avglen;
Output:
N  avg    calc   %diff
1 1.0000 1.0000  0.000
2 1.5000 1.5000  0.003
3 1.8886 1.8889  0.015
4 2.2180 2.2188  0.033
5 2.5104 2.5104  0.000
6 2.7745 2.7747  0.006
7 3.0191 3.0181  0.033
8 3.2433 3.2450  0.052
9 3.4583 3.4583  0.001
10 3.6597 3.6602  0.015
11 3.8524 3.8524  0.001
12 4.0352 4.0361  0.022
13 4.2147 4.2123  0.055
14 4.3853 4.3820  0.075
15 4.5453 4.5458  0.011
16 4.7055 4.7043  0.027
17 4.8592 4.8579  0.028
18 5.0062 5.0071  0.017
19 5.1535 5.1522  0.025
20 5.2955 5.2936  0.035

## BBC BASIC

@% = &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)
Output:
1             1.0000    1.0000    0.0000%
2             1.4995    1.5000   -0.0366%
3             1.8879    1.8889   -0.0509%
4             2.2193    2.2188    0.0240%
5             2.5105    2.5104    0.0057%
6             2.7755    2.7747    0.0293%
7             3.0199    3.0181    0.0573%
8             3.2396    3.2450   -0.1664%
9             3.4562    3.4583   -0.0609%
10            3.6578    3.6602   -0.0659%
11            3.8523    3.8524   -0.0025%
12            4.0336    4.0361   -0.0602%
13            4.2139    4.2123    0.0366%
14            4.3816    4.3820   -0.0105%
15            4.5432    4.5458   -0.0570%
16            4.7108    4.7043    0.1386%
17            4.8578    4.8579   -0.0018%
18            5.0063    5.0071   -0.0144%
19            5.1564    5.1522    0.0814%
20            5.2945    5.2936    0.0166%

## C

#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <time.h>

#define MAX_N 20
#define TIMES 1000000

double factorial(int n) {
double f = 1;
int i;
for (i = 1; i <= n; i++) f *= i;
return f;
}

double expected(int n) {
double sum = 0;
int i;
for (i = 1; i <= n; i++)
sum += factorial(n) / pow(n, i) / factorial(n - i);
return sum;
}

int randint(int n) {
int r, rmax = RAND_MAX / n * n;
while ((r = rand()) >= rmax);
return r / (RAND_MAX / n);
}

int test(int n, int times) {
int i, count = 0;
for (i = 0; i < times; i++) {
int x = 1, bits = 0;
while (!(bits & x)) {
count++;
bits |= x;
x = 1 << randint(n);
}
}
return count;
}

int main(void) {
srand(time(0));
puts(" n\tavg\texp.\tdiff\n-------------------------------");

int n;
for (n = 1; n <= MAX_N; n++) {
int cnt = test(n, TIMES);
double avg = (double)cnt / TIMES;
double theory = expected(n);
double diff = (avg / theory - 1) * 100;
printf("%2d %8.4f %8.4f %6.3f%%\n", n, avg, theory, diff);
}
return 0;
}
Output:
n      avg     exp.    diff
-------------------------------
1   1.0000   1.0000  0.000%
2   1.4998   1.5000 -0.015%
3   1.8879   1.8889 -0.051%
4   2.2181   2.2188 -0.029%
5   2.5107   2.5104  0.012%
6   2.7741   2.7747 -0.021%
7   3.0168   3.0181 -0.044%
8   3.2455   3.2450  0.014%
9   3.4591   3.4583  0.023%
10   3.6596   3.6602 -0.017%
11   3.8519   3.8524 -0.013%
12   4.0384   4.0361  0.059%
13   4.2106   4.2123 -0.042%
14   4.3840   4.3820  0.044%
15   4.5449   4.5458 -0.020%
16   4.7058   4.7043  0.033%
17   4.8549   4.8579 -0.060%
18   5.0084   5.0071  0.026%
19   5.1479   5.1522 -0.084%
20   5.2957   5.2936  0.040%

## C++

Partial translation of C using stl and std.

#include <random>
#include <vector>
#include <iostream>

#define MAX_N 20
#define TIMES 1000000

/**
* Used to generate a uniform random distribution
*/

static std::random_device rd; //Will be used to obtain a seed for the random number engine
static std::mt19937 gen(rd()); //Standard mersenne_twister_engine seeded with rd()
static std::uniform_int_distribution<> dis;

int randint(int n) {
int r, rmax = RAND_MAX / n * n;
dis=std::uniform_int_distribution(0,rmax) ;
r = dis(gen);
return r / (RAND_MAX / n);
}

unsigned long factorial(size_t n) {
//Factorial using dynamic programming to memoize the values.
static std::vector<unsigned long>factorials{1,1,2};
for (;factorials.size() <= n;)
factorials.push_back(factorials.back()*factorials.size());
return factorials[n];
}

long double expected(size_t n) {
long double sum = 0;
for (size_t i = 1; i <= n; i++)
sum += factorial(n) / pow(n, i) / factorial(n - i);
return sum;
}

int test(int n, int times) {
int i, count = 0;
for (i = 0; i < times; i++) {
unsigned int x = 1, bits = 0;
while (!(bits & x)) {
count++;
bits |= x;
x = static_cast<unsigned int>(1 << randint(n));
}
}
return count;
}

int main() {
puts(" n\tavg\texp.\tdiff\n-------------------------------");

int n;
for (n = 1; n <= MAX_N; n++) {
int cnt = test(n, TIMES);
long double avg = (double)cnt / TIMES;
long double theory = expected(static_cast<size_t>(n));
long double diff = (avg / theory - 1) * 100;
printf("%2d %8.4f %8.4f %6.3f%%\n", n, static_cast<double>(avg), static_cast<double>(theory), static_cast<double>(diff));
}
return 0;
}

Output:
n      avg     exp.    diff
-------------------------------
1   1.0000   1.0000  0.000%
2   1.4998   1.5000 -0.016%
3   1.8883   1.8889 -0.032%
4   2.2188   2.2188  0.000%
5   2.5105   2.5104  0.004%
6   2.7760   2.7747  0.047%
7   3.0180   3.0181 -0.004%
8   3.2448   3.2450 -0.007%
9   3.4580   3.4583 -0.010%
10   3.6614   3.6602  0.032%
11   3.8532   3.8524  0.022%
12   4.0349   4.0361 -0.029%
13   4.2153   4.2123  0.070%
14   4.3819   4.3820 -0.003%
15   4.5494   4.5458  0.079%
16   4.7082   4.7043  0.084%
17   4.8576   4.8579 -0.005%
18   5.0028   5.0071 -0.084%
19   5.1484   5.1522 -0.073%
20   5.2939   5.2936  0.006%

## Clojure

Translation of: Python
(ns cyclelengths
(:gen-class))

(defn factorial [n]
" n! "
(apply *' (range 1 (inc n)))) ; Use *' (vs. *) to allow arbitrary length arithmetic

(defn pow [n i]
" n^i"
(apply *' (repeat i n)))

(defn analytical [n]
" Analytical Computation "
(->>(range 1 (inc n))
(map #(/ (factorial n) (pow n %) (factorial (- n %)))) ;calc n %))
(reduce + 0)))

;; Number of random times to test each n
(def TIMES 1000000)

(defn single-test-cycle-length [n]
" Single random test of cycle length "
(loop [count 0
bits 0
x 1]
(if (zero? (bit-and x bits))
(recur (inc count) (bit-or bits x) (bit-shift-left 1 (rand-int n)))
count)))

(defn avg-cycle-length [n times]
" Average results of single tests of cycle lengths "
(/
(reduce +
(for [i (range times)]
(single-test-cycle-length n)))
times))

;; Show Results
(println "\tAvg\t\tExp\t\tDiff")
(doseq [q (range 1 21)
:let [anal (double (analytical q))
avg (double (avg-cycle-length q TIMES))
diff (Math/abs (* 100 (- 1 (/ avg anal))))]]
(println (format "%3d\t%.4f\t%.4f\t%.2f%%" q avg anal diff)))

Output:
Avg	Exp	Diff
1	1.0000	1.0000	0.00%
2	1.4995	1.5000	0.03%
3	1.8899	1.8889	0.05%
4	2.2178	2.2188	0.04%
5	2.5118	2.5104	0.06%
6	2.7773	2.7747	0.09%
7	3.0177	3.0181	0.02%
8	3.2448	3.2450	0.01%
9	3.4587	3.4583	0.01%
10	3.6594	3.6602	0.02%
11	3.8553	3.8524	0.08%
12	4.0335	4.0361	0.06%
13	4.2113	4.2123	0.03%
14	4.3823	4.3820	0.01%
15	4.5491	4.5458	0.07%
16	4.7035	4.7043	0.02%
17	4.8580	4.8579	0.00%
18	5.0050	5.0071	0.04%
19	5.1543	5.1522	0.04%
20	5.2956	5.2936	0.04%

## D

Translation of: Perl 6
import std.stdio, std.random, std.math, std.algorithm, std.range, std.format;

real analytical(in int n) pure nothrow @safe /*@nogc*/ {
enum aux = (int k) => reduce!q{a * b}(1.0L, iota(n - k + 1, n + 1));
return iota(1, n + 1)
.map!(k => (aux(k) * k ^^ 2) / (real(n) ^^ (k + 1)))
.sum;
}

size_t loopLength(size_t maxN)(in int size, ref Xorshift rng) {
__gshared static bool[maxN + 1] seen;
seen[0 .. size + 1] = false;
int current = 1;
size_t steps = 0;
while (!seen[current]) {
seen[current] = true;
current = uniform(1, size + 1, rng);
steps++;
}
return steps;
}

void main() {
enum maxN = 40;
enum nTrials = 300_000;
auto rng = Xorshift(unpredictableSeed);
writeln(" n average analytical (error)");
writeln("=== ========= ============ ==========");

foreach (immutable n; 1 .. maxN + 1) {
long total = 0;
foreach (immutable _; 0 .. nTrials)
total += loopLength!maxN(n, rng);
immutable average = total / real(nTrials);
immutable an = n.analytical;
immutable percentError = abs(an - average) / an * 100;
immutable errorS = format("%2.4f", percentError);
writefln("%3d  %9.5f  %12.5f (%7s%%)",
n, average, an, errorS);
}
}
Output:
n    average    analytical     (error)
===  =========  ============  ==========
1    1.00000       1.00000  ( 0.0000%)
2    1.50017       1.50000  ( 0.0111%)
3    1.88932       1.88889  ( 0.0226%)
4    2.21795       2.21875  ( 0.0362%)
5    2.51159       2.51040  ( 0.0474%)
6    2.77373       2.77469  ( 0.0345%)
7    3.01894       3.01814  ( 0.0264%)
8    3.24734       3.24502  ( 0.0716%)
9    3.45876       3.45832  ( 0.0127%)
10    3.66595       3.66022  ( 0.1567%)
11    3.85000       3.85237  ( 0.0616%)
12    4.03532       4.03607  ( 0.0187%)
13    4.20879       4.21235  ( 0.0843%)
14    4.37664       4.38203  ( 0.1230%)
15    4.54986       4.54581  ( 0.0892%)
16    4.70431       4.70426  ( 0.0010%)
17    4.85640       4.85787  ( 0.0302%)
18    5.01359       5.00706  ( 0.1303%)
19    5.15487       5.15220  ( 0.0519%)
20    5.29486       5.29358  ( 0.0241%)
21    5.43276       5.43150  ( 0.0231%)
22    5.56570       5.56620  ( 0.0088%)
23    5.70611       5.69788  ( 0.1443%)
24    5.82618       5.82675  ( 0.0098%)
25    5.94846       5.95298  ( 0.0759%)
26    6.07440       6.07672  ( 0.0381%)
27    6.20717       6.19811  ( 0.1461%)
28    6.31546       6.31729  ( 0.0290%)
29    6.44201       6.43437  ( 0.1187%)
30    6.54592       6.54946  ( 0.0540%)
31    6.65818       6.66265  ( 0.0671%)
32    6.77215       6.77405  ( 0.0279%)
33    6.88381       6.88372  ( 0.0013%)
34    6.99790       6.99175  ( 0.0880%)
35    7.10990       7.09820  ( 0.1648%)
36    7.20391       7.20316  ( 0.0104%)
37    7.30085       7.30667  ( 0.0796%)
38    7.40366       7.40880  ( 0.0693%)
39    7.51864       7.50959  ( 0.1204%)
40    7.60255       7.60911  ( 0.0863%)

## EchoLisp

(lib 'math) ;; Σ aka (sigma f(n) nfrom nto)

(define (f-count N (times 100000))
(define count 0)
(for ((i times))

;; new random f mapping from 0..N-1 to 0..N-1
;; (f n) is NOT (random N)
;; because each call (f n) must return the same value

(define f (build-vector N (lambda(i) (random N))))

(define hits (make-vector N))
(define n 0)
(while (zero? [hits n])
(++ count)
(vector+= hits n 1)
(set! n [f n])))
(// count times))

(define (f-anal N)
(Σ (lambda(i) (// (! N) (! (- N i)) (^ N i))) 1 N))

(decimals 5)
(define (f-print (maxN 21))
(for ((N (in-range 1 maxN)))
(define fc (f-count N))
(define fa (f-anal N))
(printf "%3d %10d %10d  %10.2d %%" N fc fa (// (abs (- fa fc)) fc 0.01))))

Output:
(f-print)
1          1          1          0 %
2    1.49908        1.5       0.06 %
3    1.89059    1.88889       0.09 %
4    2.21709    2.21875       0.07 %
5    2.50629     2.5104       0.16 %
6    2.77027    2.77469       0.16 %
7    3.01739    3.01814       0.02 %
8    3.23934    3.24502       0.18 %
9    3.45862    3.45832       0.01 %
10    3.65959    3.66022       0.02 %
11    3.85897    3.85237       0.17 %
12    4.04188    4.03607       0.14 %
13    4.21226    4.21235          0 %
14    4.38021    4.38203       0.04 %
15    4.54158    4.54581       0.09 %
16    4.70633    4.70426       0.04 %
17    4.86109    4.85787       0.07 %
18    4.99903    5.00706       0.16 %
19    5.15873     5.1522       0.13 %
20    5.30243    5.29358       0.17 %

## Elixir

Translation of: Ruby
Works with: Elixir version 1.1+
defmodule RC do
def factorial(0), do: 1
def factorial(n), do: Enum.reduce(1..n, 1, &(&1 * &2))

def loop_length(n), do: loop_length(n, MapSet.new)

defp loop_length(n, set) do
r = :rand.uniform(n)
if r in set, do: MapSet.size(set), else: loop_length(n, MapSet.put(set, r))
end

IO.puts " N average analytical (error) "
IO.puts "=== ========= ========== ========="
Enum.each(1..20, fn n ->
avg = Enum.reduce(1..runs, 0, fn _,sum -> sum + loop_length(n) end) / runs
analytical = Enum.reduce(1..n, 0, fn i,sum ->
sum + (factorial(n) / :math.pow(n, i) / factorial(n-i))
end)
:io.format "~3w ~9.4f ~9.4f (~6.2f%)~n", [n, avg, analytical, abs(avg/analytical - 1)*100]
end)
end
end

runs = 1_000_000
Output:
N    average   analytical   (error)
===  =========  ==========  =========
1     1.0000      1.0000  (  0.00%)
2     1.5001      1.5000  (  0.00%)
3     1.8892      1.8889  (  0.02%)
4     2.2189      2.2188  (  0.01%)
5     2.5113      2.5104  (  0.04%)
6     2.7749      2.7747  (  0.01%)
7     3.0185      3.0181  (  0.01%)
8     3.2456      3.2450  (  0.02%)
9     3.4612      3.4583  (  0.08%)
10     3.6573      3.6602  (  0.08%)
11     3.8524      3.8524  (  0.00%)
12     4.0357      4.0361  (  0.01%)
13     4.2102      4.2123  (  0.05%)
14     4.3813      4.3820  (  0.02%)
15     4.5422      4.5458  (  0.08%)
16     4.7057      4.7043  (  0.03%)
17     4.8581      4.8579  (  0.01%)
18     5.0045      5.0071  (  0.05%)
19     5.1533      5.1522  (  0.02%)
20     5.2951      5.2936  (  0.03%)

## F#

Translation of: Scala

But uses the Gamma function instead of factorials.

open System

let gamma z =
let lanczosCoefficients = [76.18009172947146;-86.50532032941677;24.01409824083091;-1.231739572450155;0.1208650973866179e-2;-0.5395239384953e-5]
let rec sumCoefficients acc i coefficients =
match coefficients with
| [] -> acc
| h::t -> sumCoefficients (acc + (h/i)) (i+1.0) t
let gamma = 5.0
let x = z - 1.0
Math.Pow(x + gamma + 0.5, x + 0.5) * Math.Exp( -(x + gamma + 0.5) ) * Math.Sqrt( 2.0 * Math.PI ) * sumCoefficients 1.000000000190015 (x + 1.0) lanczosCoefficients

let factorial n = gamma ((float n) + 1.)

let expected n =
seq {for i in 1 .. n do yield (factorial n) / System.Math.Pow((float n), (float i)) / (factorial (n - i)) }
|> Seq.sum

let r = System.Random()

let trial n =
let count = ref 0
let x = ref 1
let bits = ref 0
while (!bits &&& !x) = 0 do
count := !count + 1
bits := !bits ||| !x
x := 1 <<< r.Next(n)
!count

let tested n times = (float (Seq.sum (seq { for i in 1 .. times do yield (trial n) }))) / (float times)

let results = seq {
for n in 1 .. 20 do
let avg = tested n 1000000
let theory = expected n
yield n, avg, theory
}

[<EntryPoint>]
let main argv =
printfn " N average analytical (error)"
printfn "------------------------------------"
results
|> Seq.iter (fun (n, avg, theory) ->
printfn "%2i  %2.6f  %2.6f  %+2.3f%%" n avg theory ((avg / theory - 1.) * 100.))
0

Output:
N     average   analytical   (error)
------------------------------------
1    1.000000    1.000000    +0.000%
2    1.498934    1.500000    -0.071%
3    1.889318    1.888889    +0.023%
4    2.219397    2.218750    +0.029%
5    2.510618    2.510400    +0.009%
6    2.771914    2.774691    -0.100%
7    3.014726    3.018139    -0.113%
8    3.245022    3.245018    +0.000%
9    3.457096    3.458316    -0.035%
10    3.660337    3.660216    +0.003%
11    3.849770    3.852372    -0.068%
12    4.038977    4.036074    +0.072%
13    4.213248    4.212348    +0.021%
14    4.380451    4.382029    -0.036%
15    4.541868    4.545807    -0.087%
16    4.704117    4.704258    -0.003%
17    4.858934    4.857871    +0.022%
18    5.004236    5.007063    -0.056%
19    5.154166    5.152196    +0.038%
20    5.298119    5.293585    +0.086%

## Go

package main

import (
"fmt"
"math"
"math/rand"
)

const nmax = 20

func main() {
fmt.Println(" N average analytical (error)")
fmt.Println("=== ========= ============ =========")
for n := 1; n <= nmax; n++ {
a := avg(n)
b := ana(n)
fmt.Printf("%3d  %9.4f  %12.4f (%6.2f%%)\n",
n, a, b, math.Abs(a-b)/b*100)
}
}

func avg(n int) float64 {
const tests = 1e4
sum := 0
for t := 0; t < tests; t++ {
var v [nmax]bool
for x := 0; !v[x]; x = rand.Intn(n) {
v[x] = true
sum++
}
}
return float64(sum) / tests
}

func ana(n int) float64 {
nn := float64(n)
term := 1.
sum := 1.
for i := nn - 1; i >= 1; i-- {
term *= i / nn
sum += term
}
return sum
}
Output:
N    average    analytical    (error)
===  =========  ============  =========
1     1.0000        1.0000  (  0.00%)
2     1.5007        1.5000  (  0.05%)
3     1.8959        1.8889  (  0.37%)
4     2.2138        2.2188  (  0.22%)
5     2.5013        2.5104  (  0.36%)
6     2.7940        2.7747  (  0.70%)
7     3.0197        3.0181  (  0.05%)
8     3.2715        3.2450  (  0.82%)
9     3.4147        3.4583  (  1.26%)
10     3.6758        3.6602  (  0.43%)
11     3.8672        3.8524  (  0.38%)
12     4.0309        4.0361  (  0.13%)
13     4.2153        4.2123  (  0.07%)
14     4.3380        4.3820  (  1.00%)
15     4.5030        4.5458  (  0.94%)
16     4.7563        4.7043  (  1.11%)
17     4.8616        4.8579  (  0.08%)
18     4.9933        5.0071  (  0.27%)
19     5.1534        5.1522  (  0.02%)
20     5.3031        5.2936  (  0.18%)

import System.Random
import qualified Data.Set as S
import Text.Printf

findRep :: (Random a, Integral a, RandomGen b) => a -> b -> (a, b)
findRep n gen = findRep' (S.singleton 1) 1 gen
where
findRep'
seen len gen'
| S.member fx seen = (len, gen'
')
| otherwise = findRep'
(S.insert fx seen) (len + 1) gen''
where
(fx, gen'') = randomR (1, n) gen'

statistical :: (Integral a, Random b, Integral b, RandomGen c, Fractional d) =>
a -> b -> c -> (d, c)
statistical samples size gen =
let (total, gen'
) = sar samples gen 0
in ((fromIntegral total) / (fromIntegral samples), gen')
where
sar 0 gen'
acc = (acc, gen')
sar samples'
gen' acc =
let (len, gen'
') = findRep size gen'
in sar (samples' - 1) gen'' (acc + len)

factorial :: (Integral a) => a -> a
factorial n = foldl (*) 1 [1..n]

analytical :: (Integral a, Fractional b) => a -> b
analytical n = sum [fromIntegral num /
fromIntegral (factorial (n - i)) /
fromIntegral (n ^ i) |
i <- [1..n]]
where num = factorial n

test :: (Integral a, Random b, Integral b, PrintfArg b, RandomGen c) =>
a -> [b] -> c -> IO c
test _ [] gen = return gen
test samples (x:xs) gen = do
let (st, gen'
) = statistical samples x gen
an = analytical x
err = abs (st - an) / st * 100.0
str = printf "%3d  %9.4f  %12.4f (%6.2f%%)\n"
x (st :: Float) (an :: Float) (err :: Float)
putStr str
test samples xs gen'

main :: IO ()
main = do
putStrLn " N average analytical (error)"
putStrLn "=== ========= ============ ========="
let samples = 10000 :: Integer
range = [1..20] :: [Integer]
_ <- test samples range \$ mkStdGen 0
return ()
N    average    analytical    (error)
===  =========  ============  =========
1     1.0000        1.0000  (  0.00%)
2     1.4941        1.5000  (  0.39%)
3     1.8895        1.8889  (  0.03%)
4     2.2246        2.2188  (  0.26%)
5     2.5158        2.5104  (  0.21%)
6     2.7875        2.7747  (  0.46%)
7     3.0425        3.0181  (  0.80%)
8     3.2157        3.2450  (  0.91%)
9     3.4534        3.4583  (  0.14%)
10     3.6561        3.6602  (  0.11%)
11     3.8357        3.8524  (  0.43%)
12     4.0291        4.0361  (  0.17%)
13     4.1819        4.2123  (  0.73%)
14     4.3469        4.3820  (  0.81%)
15     4.4942        4.5458  (  1.15%)
16     4.7093        4.7043  (  0.11%)
17     4.8288        4.8579  (  0.60%)
18     5.0021        5.0071  (  0.10%)
19     5.1980        5.1522  (  0.88%)
20     5.2961        5.2936  (  0.05%)

## J

First, let's consider an exact, brute force approach.

Since J array indices start at 0, we'll work with 0..N-1 instead of 1..N, dealing with the difference at the boundaries.

We can implement f as {&LIST where LIST is an arbitrary list of N numbers, each picked independently from the range 0..(N-1). We can incrementally build the described sequence using (, [email protected]{:) - here we extend the sequence by applying f to the last element of the sequence. Since we are only concerned with the sequence up to the point of the first repeat, we can select the unique values giving us ([email protected], [email protected]{:). This routine stops changing when we reach the desired length, so we can repeatedly apply it forever. For example:

([email protected], {&0 0@{:)^:_] 0
0
([email protected], {&0 0@{:)^:_] 1
1 0

Once we have the sequence, we can count how many elements are in it.

0 0 ([: # (] [email protected], {:@] { [)^:_) 1
2

Meanwhile, we can also generate all possible values of 1..N by counting out N^N values and breaking out the result as a base N list of digits.

(#.inv [email protected]^~)2
0 0
0 1
1 0
1 1

All that's left is to count the lengths of all possible sequences for all possible distinct instances of f and average the results:

(+/ % #)@,@((#.inv [email protected]^~) ([: # (] [email protected], {:@] { [)^:_)"1 0/ i.)1
1
(+/ % #)@,@((#.inv [email protected]^~) ([: # (] [email protected], {:@] { [)^:_)"1 0/ i.)2
1.5
(+/ % #)@,@((#.inv [email protected]^~) ([: # (] [email protected], {:@] { [)^:_)"1 0/ i.)3
1.88889
(+/ % #)@,@((#.inv [email protected]^~) ([: # (] [email protected], {:@] { [)^:_)"1 0/ i.)4
2.21875
(+/ % #)@,@((#.inv [email protected]^~) ([: # (] [email protected], {:@] { [)^:_)"1 0/ i.)5
2.5104
(+/ % #)@,@((#.inv [email protected]^~) ([: # (] [email protected], {:@] { [)^:_)"1 0/ i.)6
2.77469

Meanwhile the analytic solution (derived by reading the Ada implementation) looks like this:

ana=: +/@([email protected][ % [email protected] * ^) 1+i.
ana"0]1 2 3 4 5 6
1 1.5 1.88889 2.21875 2.5104 2.77469

To get our simulation, we can take the exact approach and replace the part that generates all possible values for f with a random mechanism. Since the task does not specify how long to run the simulation, and to make this change easy, we'll use N*1e4 tests.

sim=: (+/ % #)@,@((][email protected]\$~1e4,]) ([: # (] [email protected], {:@] { [)^:_)"1 0/ i.)
sim"0]1 2 3 4 5 6
1 1.5034 1.8825 2.22447 2.51298 2.76898

The simulation approach is noticeably slower than the analytic approach, while being less accurate.

Finally, we can generate our desired results:

(;:'N average analytic error'),:,.each(;ana"0 ([;];-|@%[) sim"0)1+i.20
+--+-------+--------+-----------+
|N |average|analytic|error |
+--+-------+--------+-----------+
| 1| 1| 1 | 0|
| 2| 1.5|1.49955 | 0.0003|
| 3|1.88889| 1.8928 | 0.00207059|
| 4|2.21875|2.23082 | 0.00544225|
| 5| 2.5104|2.52146 | 0.00440567|
| 6|2.77469|2.78147 | 0.00244182|
| 7|3.01814| 3.0101 | 0.00266346|
| 8|3.24502|3.25931 | 0.00440506|
| 9|3.45832|3.45314 | 0.00149532|
|10|3.66022| 3.6708 | 0.00289172|
|11|3.85237|3.84139 | 0.00285049|
|12|4.03607|4.03252 |0.000881304|
|13|4.21235|4.18358 | 0.00682833|
|14|4.38203|4.38791 | 0.00134132|
|15|4.54581|4.54443 |0.000302246|
|16|4.70426|4.71351 | 0.00196721|
|17|4.85787|4.85838 |0.000104089|
|18|5.00706|5.00889 |0.000365752|
|19| 5.1522|5.14785 |0.000843052|
|20|5.29358|5.28587 | 0.00145829|
+--+-------+--------+-----------+

Here, error is the difference between the two values divided by the analytic value.

## Java

This uses a 0-based index (0, 1, ..., n-1) as opposed to the 1-based index (1, 2, ..., n) specified in the question, because it fits better with the native structure of Java.

import java.util.HashSet;
import java.util.Random;
import java.util.Set;

public class AverageLoopLength {

private static final int N = 100000;

//analytical(n) = sum_(i=1)^n (n!/(n-i)!/n**i)
private static double analytical(int n) {
double[] factorial = new double[n + 1];
double[] powers = new double[n + 1];
powers[0] = 1.0;
factorial[0] = 1.0;
for (int i = 1; i <= n; i++) {
factorial[i] = factorial[i - 1] * i;
powers[i] = powers[i - 1] * n;
}
double sum = 0;
//memoized factorial and powers
for (int i = 1; i <= n; i++) {
sum += factorial[n] / factorial[n - i] / powers[i];
}
return sum;
}

private static double average(int n) {
Random rnd = new Random();
double sum = 0.0;
for (int a = 0; a < N; a++) {
int[] random = new int[n];
for (int i = 0; i < n; i++) {
random[i] = rnd.nextInt(n);
}
Set<Integer> seen = new HashSet<>(n);
int current = 0;
int length = 0;
length++;
current = random[current];
}
sum += length;
}
return sum / N;
}

public static void main(String[] args) {
System.out.println(" N average analytical (error)");
System.out.println("=== ========= ============ =========");
for (int i = 1; i <= 20; i++) {
double avg = average(i);
double ana = analytical(i);
System.out.println(String.format("%3d  %9.4f  %12.4f (%6.2f%%)", i, avg, ana, ((ana - avg) / ana * 100)));
}
}
}

## Julia

Translation of: Python
# Version 5.2
analytical(n::Integer) = sum(factorial(n) / big(n) ^ i / factorial(n - i) for i = 1:n)

function test(n::Integer, times::Integer = 1000000)
c = 0
for i = range(0, times)
x, bits = 1, 0
while (bits & x) == 0
c += 1
bits |= x
x = 1 << rand(0:(n - 1))
end
end
return c / times
end

function main(n::Integer)
println(" n\tavg\texp.\tdiff\n-------------------------------")
for n in 1:n
avg = test(n)
theory = analytical(n)
diff = (avg / theory - 1) * 100
@printf(STDOUT, "%2d %8.4f %8.4f %6.3f%%\n", n, avg, theory, diff)
end
end

main(20)

Output:
n	avg	exp.	diff
-------------------------------
1   1.0000   1.0000  0.000%
2   1.4998   1.5000 -0.015%
3   1.8895   1.8889  0.034%
4   2.2171   2.2188 -0.075%
5   2.5082   2.5104 -0.088%
6   2.7729   2.7747 -0.063%
7   3.0171   3.0181 -0.033%
8   3.2439   3.2450 -0.034%
9   3.4578   3.4583 -0.016%
10   3.6616   3.6602  0.038%
11   3.8525   3.8524  0.004%
12   4.0353   4.0361 -0.020%
13   4.2126   4.2123  0.006%
14   4.3835   4.3820  0.034%
15   4.5428   4.5458 -0.067%
16   4.7027   4.7043 -0.033%
17   4.8560   4.8579 -0.039%
18   5.0054   5.0071 -0.033%
19   5.1492   5.1522 -0.058%
20   5.2896   5.2936 -0.076%

## Kotlin

Translation of: Go
const val NMAX  = 20
const val TESTS = 1000000
val rand = java.util.Random()

fun avg(n: Int): Double {
var sum = 0
for (t in 0 until TESTS) {
val v = BooleanArray(NMAX)
var x = 0
while (!v[x]) {
v[x] = true
sum++
x = rand.nextInt(n)
}
}
return sum.toDouble() / TESTS
}

fun ana(n: Int): Double {
val nn = n.toDouble()
var term = 1.0
var sum = 1.0
for (i in n - 1 downTo 1) {
term *= i / nn
sum += term
}
return sum
}

fun main(args: Array<String>) {
println(" N average analytical (error)")
println("=== ========= ============ =========")
for (n in 1..NMAX) {
val a = avg(n)
val b = ana(n)
println(String.format("%3d  %6.4f  %10.4f (%4.2f%%)", n, a, b, Math.abs(a - b) / b * 100.0))
}
}

Sample output:

Output:
N    average    analytical    (error)
===  =========  ============  =========
1   1.0000       1.0000      (0.00%)
2   1.5004       1.5000      (0.03%)
3   1.8890       1.8889      (0.00%)
4   2.2179       2.2188      (0.04%)
5   2.5108       2.5104      (0.02%)
6   2.7738       2.7747      (0.03%)
7   3.0178       3.0181      (0.01%)
8   3.2482       3.2450      (0.10%)
9   3.4572       3.4583      (0.03%)
10   3.6608       3.6602      (0.02%)
11   3.8545       3.8524      (0.06%)
12   4.0378       4.0361      (0.04%)
13   4.2131       4.2123      (0.02%)
14   4.3795       4.3820      (0.06%)
15   4.5481       4.5458      (0.05%)
16   4.7044       4.7043      (0.00%)
17   4.8610       4.8579      (0.06%)
18   5.0027       5.0071      (0.09%)
19   5.1498       5.1522      (0.05%)
20   5.2941       5.2936      (0.01%)

## Liberty BASIC

Translation of: BBC BASIC

MAXN = 20
TIMES = 10000'00

't0=time\$("ms")
FOR n = 1 TO MAXN
avg = FNtest(n, TIMES)
theory = FNanalytical(n)
diff = (avg / theory - 1) * 100
PRINT n, avg, theory, using("##.####",diff); "%"
NEXT
't1=time\$("ms")
'print t1-t0; " ms"
END

function FNanalytical(n)
FOR i = 1 TO n
s = s+ FNfactorial(n) / n^i / FNfactorial(n-i)
NEXT
FNanalytical = s
end function

function FNtest(n, times)
FOR i = 1 TO times
x = 1 : b = 0
WHILE (b AND x) = 0
c = c + 1
b = b OR x
x = 2^int(n*RND(1))
WEND
NEXT
FNtest = c / times
end function

function FNfactorial(n)
IF n=1 OR n=0 THEN FNfactorial=1 ELSE FNfactorial= n * FNfactorial(n-1)
end function

Output:
1             1             1              0.0000%
2             1.4759        1.5           -1.6067%
3             1.8868        1.88888889    -0.1106%
4             2.2139        2.21875       -0.2186%
5             2.4784        2.5104        -1.2747%
6             2.7888        2.77469136     0.5085%
7             2.9846        3.0181387     -1.1112%
8             3.2645        3.24501801     0.6004%
9             3.464         3.45831574     0.1644%
10            3.6602        3.66021568    -0.0004%
11            3.8255        3.85237205    -0.6975%
12            4.019         4.03607368    -0.4230%
13            4.2033        4.21234791    -0.2148%
14            4.3985        4.38202942     0.3759%
15            4.5868        4.54580729     0.9018%
16            4.6705        4.70425825    -0.7176%
17            4.8807        4.85787082     0.4699%
18            4.9759        5.0070631     -0.6224%
19            5.1755        5.1521962      0.4523%
20            5.2792        5.29358459    -0.2717%

## Mathematica / Wolfram Language

[email protected][
Table[{n, #[[1]], #[[2]],
Row[{Round[10000 Abs[#[[1]] - #[[2]]]/#[[2]]]/100., "%"}]} &@
N[{Mean[Array[
[email protected][#, 1, UnsameQ[##] &, All] - 1 &[# /.
MapIndexed[#2[[1]] -> #1 &,
RandomInteger[{1, n}, n]] &] &, 10000]],
Sum[n! n^(n - k - 1)/(n - k)!, {k, n}]/n^(n - 1)}, 5], {n, 1,
20}], {"N", "average", "analytical", "error"}]
Output:
N	average	analytical	error
1	1.0000	1.0000		0.%
2	1.5017	1.5000		0.11%
3	1.8910	1.8889		0.11%
4	2.2334	2.2188		0.66%
5	2.5090	2.5104		0.06%
6	2.8092	2.7747		1.24%
7	3.0468	3.0181		0.95%
8	3.2253	3.2450		0.61%
9	3.4695	3.4583		0.32%
10	3.6661	3.6602		0.16%
11	3.8662	3.8524		0.36%
12	4.0393	4.0361		0.08%
13	4.2232	4.2123		0.26%
14	4.3496	4.3820		0.74%
15	4.5706	4.5458		0.55%
16	4.6963	4.7043		0.17%
17	4.8548	4.8579		0.06%
18	5.0671	5.0071		1.2%
19	5.1702	5.1522		0.35%
20	5.2264	5.2936		1.27%

## Nim

Translation of: C
import random, math, strfmt
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 random(n)

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
printlnfmt "{:2} {:8.4f} {:8.4f} {:6.3f}%", n, avg, theory, diff
Output:
n	avg	exp.	diff
-------------------------------
1   1.0000   1.0000      0%
2   1.5001   1.5000  0.008%
3   1.8884   1.8889 -0.025%
4   2.2187   2.2187 -0.000%
5   2.5098   2.5104 -0.025%
6   2.7752   2.7747  0.017%
7   3.0175   3.0181 -0.020%
8   3.2411   3.2450 -0.120%
9   3.4565   3.4583 -0.054%
10   3.6599   3.6602 -0.010%
11   3.8555   3.8524  0.081%
12   4.0381   4.0361  0.051%
13   4.2124   4.2123  0.000%
14   4.3813   4.3820 -0.017%
15   4.5471   4.5458  0.027%
16   4.7009   4.7043 -0.072%
17   4.8589   4.8579  0.021%
18   5.0054   5.0071 -0.034%
19   5.1554   5.1522  0.061%
20   5.2915   5.2936 -0.040%

## Oberon-2

MODULE AvgLoopLen;
(* Oxford Oberon-2 *)
IMPORT Random, Out;

PROCEDURE Fac(n: INTEGER; f: REAL): REAL;
BEGIN
IF n = 0 THEN
RETURN f
ELSE
RETURN Fac(n - 1,n*f)
END
END Fac;

PROCEDURE Power(n,i: INTEGER): REAL;
VAR
p: REAL;
BEGIN
p := 1.0;
WHILE i > 0 DO p := p * n; DEC(i) END;
RETURN p
END Power;

PROCEDURE Abs(x: REAL): REAL;
BEGIN
IF x < 0 THEN RETURN -x ELSE RETURN x END
END Abs;

PROCEDURE Analytical(n: INTEGER): REAL;
VAR
i: INTEGER;
res: REAL;
BEGIN
res := 0.0;
FOR i := 1 TO n DO
res := res + (Fac(n,1.0) / Power(n,i) / Fac(n - i,1.0));
END;
RETURN res
END Analytical;

PROCEDURE Averages(n: INTEGER): REAL;
CONST
times = 100000;
VAR
rnds: SET;
r,count,i: INTEGER;
BEGIN
count := 0; i := 0;
WHILE i < times DO
rnds := {};
LOOP
r := Random.Roll(n);
IF r IN rnds THEN EXIT ELSE INCL(rnds,r); INC(count) END
END;
INC(i)
END;

RETURN count / times
END Averages;

VAR
i: INTEGER;
av,an,df: REAL;
BEGIN
Random.Randomize;
Out.String(" Averages Analytical Diff% ");Out.Ln;
FOR i := 1 TO 20 DO
Out.Int(i,3); Out.String(": ");
av := Averages(i);an := Analytical(i);df := Abs(av - an) / an * 100.0;
Out.Fixed(av,10,4);Out.Fixed(an,11,4);Out.Fixed(df,10,4);Out.Ln
END
END AvgLoopLen.

Output:
Averages  Analytical  Diff%
1:     1.0000     1.0000    0.0000
2:     1.5015     1.5000    0.0993
3:     1.8868     1.8889    0.1085
4:     2.2187     2.2188    0.0005
5:     2.5119     2.5104    0.0578
6:     2.7785     2.7747    0.1366
7:     3.0184     3.0181    0.0090
8:     3.2435     3.2450    0.0471
9:     3.4585     3.4583    0.0056
10:     3.6549     3.6602    0.1463
11:     3.8559     3.8524    0.0918
12:     4.0452     4.0361    0.2264
13:     4.2097     4.2123    0.0628
14:     4.3740     4.3820    0.1830
15:     4.5583     4.5458    0.2739
16:     4.7001     4.7043    0.0882
17:     4.8654     4.8579    0.1556
18:     5.0157     5.0071    0.1731
19:     5.1515     5.1522    0.0135
20:     5.2930     5.2936    0.0105

## PARI/GP

Translation of: C
expected(n)=sum(i=1,n,n!/(n-i)!/n^i,0.);
test(n, times)={
my(ct);
for(i=1,times,
my(x=1,bits);
while(!bitand(bits,x),ct++; bits=bitor(bits,x); x = 1<<random(n))
);
ct
};
TIMES=1000000;
{for(n=1,20,
my(cnt=test(n, TIMES),avg=cnt/TIMES,ex=expected(n),diff=(avg/ex-1)*100.);
print(n"\t"avg*1."\t"ex*1."\t"diff);
)}
Output:
1       1.0000  1.0000  0.E-7
2       1.4998  1.5000  -0.012933
3       1.8891  1.8889  0.013559
4       2.2198  2.2188  0.047369
5       2.5095  2.5104  -0.034616
6       2.7744  2.7747  -0.010248
7       3.0177  3.0181  -0.012945
8       3.2467  3.2450  0.050600
9       3.4611  3.4583  0.080278
10      3.6595  3.6602  -0.018651
11      3.8541  3.8524  0.044880
12      4.0428  4.0361  0.16690
13      4.2116  4.2123  -0.017921
14      4.3825  4.3820  0.011150
15      4.5467  4.5458  0.020562
16      4.7087  4.7043  0.095058
17      4.8573  4.8579  -0.011997
18      5.0080  5.0071  0.018312
19      5.1530  5.1522  0.015970
20      5.2970  5.2936  0.065143

## Perl

use List::Util qw(sum reduce);

sub find_loop {
my(\$n) = @_;
my(\$r,@seen);
while () { \$seen[\$r] = \$seen[(\$r = int(1+rand \$n))] ? return sum @seen : 1 }
}

print " N empiric theoric (error)\n";
print "=== ========= ============ =========\n";

my \$MAX = 20;
my \$TRIALS = 1000;

for my \$n (1 .. \$MAX) {
my \$empiric = ( sum map { find_loop(\$n) } 1..\$TRIALS ) / \$TRIALS;
my \$theoric = sum map { (reduce { \$a*\$b } \$_**2, (\$n-\$_+1)..\$n ) / \$n ** (\$_+1) } 1..\$n;

printf "%3d  %9.4f  %12.4f (%5.2f%%)\n",
\$n, \$empiric, \$theoric, 100 * (\$empiric - \$theoric) / \$theoric;
}
Output:
N    empiric      theoric     (error)
===  =========  ============  =========
1     1.0000        1.0000   ( 0.00%)
2     1.4950        1.5000   (-0.33%)
3     1.9190        1.8889   ( 1.59%)
4     2.2400        2.2188   ( 0.96%)
5     2.5120        2.5104   ( 0.06%)
6     2.7500        2.7747   (-0.89%)
7     3.0360        3.0181   ( 0.59%)
8     3.2600        3.2450   ( 0.46%)
9     3.4440        3.4583   (-0.41%)
10     3.6670        3.6602   ( 0.19%)
11     3.8340        3.8524   (-0.48%)
12     4.0450        4.0361   ( 0.22%)
13     4.2160        4.2123   ( 0.09%)
14     4.4420        4.3820   ( 1.37%)
15     4.5600        4.5458   ( 0.31%)
16     4.7940        4.7043   ( 1.91%)
17     4.7830        4.8579   (-1.54%)
18     4.9140        5.0071   (-1.86%)
19     5.2490        5.1522   ( 1.88%)
20     5.2930        5.2936   (-0.01%)

## Perl 6

Works with: rakudo version 2016.08
constant MAX_N  = 20;
constant TRIALS = 100;

for 1 .. MAX_N -> \$N {
my \$empiric = TRIALS R/ [+] find-loop(random-mapping(\$N)).elems xx TRIALS;
my \$theoric = [+]
map -> \$k { \$N ** (\$k + 1) R/ [*] flat \$k**2, \$N - \$k + 1 .. \$N }, 1 .. \$N;

FIRST say " N empiric theoric (error)";
FIRST say "=== ========= ============ =========";

printf "%3d  %9.4f  %12.4f (%4.2f%%)\n",
\$N, \$empiric,
\$theoric, 100 * abs(\$theoric - \$empiric) / \$theoric;
}

sub random-mapping { hash .list Z=> .roll given ^\$^size }
sub find-loop { 0, | %^mapping{*} ...^ { (%){\$_}++ } }
Example:
N    empiric      theoric      (error)
===  =========  ============  =========
1     1.0000        1.0000    (0.00%)
2     1.5600        1.5000    (4.00%)
3     1.7800        1.8889    (5.76%)
4     2.1800        2.2188    (1.75%)
5     2.6200        2.5104    (4.37%)
6     2.8300        2.7747    (1.99%)
7     3.1200        3.0181    (3.37%)
8     3.1400        3.2450    (3.24%)
9     3.4500        3.4583    (0.24%)
10     3.6700        3.6602    (0.27%)
11     3.8300        3.8524    (0.58%)
12     4.3600        4.0361    (8.03%)
13     3.9000        4.2123    (7.42%)
14     4.4900        4.3820    (2.46%)
15     4.9500        4.5458    (8.89%)
16     4.9800        4.7043    (5.86%)
17     4.9100        4.8579    (1.07%)
18     4.9700        5.0071    (0.74%)
19     5.1000        5.1522    (1.01%)
20     5.2300        5.2936    (1.20%)

## Phix

constant MAX = 20,
ITER = 1000000

function expected(integer n)
atom sum = 0
for i=1 to n do
sum += factorial(n) / power(n,i) / factorial(n-i)
end for
return sum
end function

function test(integer n)
integer count = 0, x, bits
for i=1 to ITER do
x = 1
bits = 0
while not and_bits(bits,x) do
count += 1
bits = or_bits(bits,x)
x = power(2,rand(n)-1)
end while
end for
return count/ITER
end function

atom av, ex
puts(1," n avg. exp. (error%)\n");
puts(1,"== ====== ====== ========\n");
for n=1 to MAX do
av = test(n)
ex = expected(n)
printf(1,"%2d %8.4f %8.4f (%5.3f%%)\n", {n,av,ex,abs(1-av/ex)*100})
end for
Output:
n     avg.     exp.  (error%)
==   ======   ======  ========
1   1.0000   1.0000  (0.000%)
2   1.5003   1.5000  (0.018%)
3   1.8880   1.8889  (0.046%)
4   2.2176   2.2188  (0.052%)
5   2.5104   2.5104  (0.001%)
6   2.7734   2.7747  (0.046%)
7   3.0198   3.0181  (0.055%)
8   3.2464   3.2450  (0.042%)
9   3.4562   3.4583  (0.062%)
10   3.6618   3.6602  (0.043%)
11   3.8511   3.8524  (0.033%)
12   4.0357   4.0361  (0.009%)
13   4.2158   4.2123  (0.083%)
14   4.3843   4.3820  (0.052%)
15   4.5410   4.5458  (0.105%)
16   4.7084   4.7043  (0.087%)
17   4.8603   4.8579  (0.049%)
18   5.0044   5.0071  (0.052%)
19   5.1516   5.1522  (0.011%)
20   5.2955   5.2936  (0.037%)

## PicoLisp

Translation of: Python
(scl 4)
(seed (in "/dev/urandom" (rd 8)))

(de fact (N)
(if (=0 N) 1 (apply * (range 1 N))) )

(de analytical (N)
(sum
'((I)
(/
(* (fact N) 1.0)
(** N I)
(fact (- N I)) ) )
(range 1 N) ) )

(de testing (N)
(let (C 0 N (dec N) X 0 B 0 I 1000000)
(do I
(zero B)
(one X)
(while (=0 (& B X))
(inc 'C)
(setq
B (| B X)
X (** 2 (rand 0 N)) ) ) )
(*/ C 1.0 I) ) )

(let F (2 8 8 6)
(tab F "N" "Avg" "Exp" "Diff")
(for I 20
(let (A (testing I) B (analytical I))
(tab F
I
(round A 4)
(round B 4)
(round
(*
(abs (- (*/ A 1.0 B) 1.0))
100 )
2 ) ) ) ) )

(bye)

## PowerShell

Works with: PowerShell version 2

function Get-AnalyticalLoopAverage ( [int]\$N )
{
# Expected loop average = sum from i = 1 to N of N! / (N-i)! / N^(N-i+1)
# Equivalently, Expected loop average = sum from i = 1 to N of F(i)
# where F(N) = 1, and F(i) = F(i+1)*i/N

\$LoopAverage = \$Fi = 1

If ( \$N -eq 1 ) { return \$LoopAverage }

ForEach ( \$i in (\$N-1)..1 )
{
\$Fi *= \$i / \$N
\$LoopAverage += \$Fi
}
return \$LoopAverage
}

function Get-ExperimentalLoopAverage ( [int]\$N, [int]\$Tests = 100000 )
{
If ( \$N -eq 1 ) { return 1 }

# Using 0 through N-1 instead of 1 through N for speed and simplicity
\$NMO = \$N - 1

# Create array to hold mapping function
\$F = New-Object int[] ( \$N )

\$Count = 0
\$Random = New-Object System.Random

ForEach ( \$Test in 1..\$Tests )
{
# Map each number to a random number
ForEach ( \$i in 0..\$NMO )
{
\$F[\$i] = \$Random.Next( \$N )
}

# For each number...
ForEach ( \$i in 0..\$NMO )
{
# Add the number to the list
\$List = @()
\$Count++
\$List += \$X = \$i

# If loop does not yet exist in list...
While ( \$F[\$X] -notin \$List )
{
# Go to the next mapped number and add it to the list
\$Count++
\$List += \$X = \$F[\$X]
}
}
}
\$LoopAvereage = \$Count / \$N / \$Tests
return \$LoopAvereage
}

Note: The use of the [pscustomobject] type accelerator to simplify making the test result table look pretty requires PowerShell 3.0.

# Display results for N = 1 through 20
ForEach ( \$N in 1..20 )
{
\$AnalyticalAverage = Get-AnalyticalLoopAverage \$N
\$ExperimentalAverage = Get-ExperimentalLoopAverage \$N
[pscustomobject] @{
N = \$N.ToString().PadLeft( 2, ' ' )
Analytical = \$AnalyticalAverage.ToString( '0.00000000' )
Experimental = \$ExperimentalAverage.ToString( '0.00000000' )
'Error (%)' = ( [math]::Abs( \$AnalyticalAverage - \$ExperimentalAverage ) / \$AnalyticalAverage * 100 ).ToString( '0.00000000' )
}
}

Output:
N  Analytical Experimental Error (%)
-  ---------- ------------ ---------
1 1.00000000 1.00000000   0.00000000
2 1.50000000 1.49985500   0.00966667
3 1.88888889 1.88713000   0.09311765
4 2.21875000 2.22103500   0.10298592
5 2.51040000 2.51069200   0.01163161
6 2.77469136 2.77264833   0.07363070
7 3.01813870 3.01547143   0.08837474
8 3.24501801 3.25003875   0.15472163
9 3.45831574 3.45067667   0.22089013
10 3.66021568 3.65659000   0.09905646
11 3.85237205 3.85669273   0.11215626
12 4.03607368 4.03813500   0.05107253
13 4.21234791 4.20946231   0.06850349
14 4.38202942 4.38458786   0.05838465
15 4.54580729 4.54466400   0.02515032
16 4.70425825 4.70146375   0.05940356
17 4.85787082 4.86807647   0.21008483
18 5.00706310 5.01939278   0.24624572
19 5.15219620 5.15179263   0.00783296
20 5.29358459 5.29214950   0.02710991

## Python

Translation of: C
from __future__ import division # Only necessary for Python 2.X
from math import factorial
from random import randrange

MAX_N = 20
TIMES = 1000000

def analytical(n):
return sum(factorial(n) / pow(n, i) / factorial(n -i) for i in range(1, n+1))

def test(n, times):
count = 0
for i in range(times):
x, bits = 1, 0
while not (bits & x):
count += 1
bits |= x
x = 1 << randrange(n)
return count / times

if __name__ == '__main__':
print(" n\tavg\texp.\tdiff\n-------------------------------")
for n in range(1, MAX_N+1):
avg = test(n, TIMES)
theory = analytical(n)
diff = (avg / theory - 1) * 100
print("%2d %8.4f %8.4f %6.3f%%" % (n, avg, theory, diff))
Output:
n	avg	exp.	diff
-------------------------------
1   1.0000   1.0000  0.000%
2   1.5006   1.5000  0.037%
3   1.8887   1.8889 -0.012%
4   2.2190   2.2188  0.011%
5   2.5101   2.5104 -0.012%
6   2.7750   2.7747  0.012%
7   3.0158   3.0181 -0.076%
8   3.2447   3.2450 -0.009%
9   3.4586   3.4583  0.009%
10   3.6598   3.6602 -0.010%
11   3.8510   3.8524 -0.036%
12   4.0368   4.0361  0.017%
13   4.2099   4.2123 -0.058%
14   4.3784   4.3820 -0.083%
15   4.5484   4.5458  0.058%
16   4.7045   4.7043  0.006%
17   4.8611   4.8579  0.067%
18   5.0074   5.0071  0.007%
19   5.1534   5.1522  0.024%
20   5.2927   5.2936 -0.017%

## R

expected <- function(size) {
result <- 0
for (i in 1:size) {
result <- result + factorial(size) / size^i / factorial(size -i)
}
result
}

knuth <- function(size) {
v <- sample(1:size, size, replace = TRUE)

visit <- vector('logical',size)
place <- 1
visit[[1]] <- TRUE
steps <- 0

repeat {
place <- v[[place]]
steps <- steps + 1
if (visit[[place]]) break
visit[[place]] <- TRUE
}
steps
}

cat(" N average analytical (error)\n")
cat("=== ========= ============ ==========\n")
for (num in 1:20) {
average <- mean(replicate(1e6, knuth(num)))
analytical <- expected(num)
error <- abs(average/analytical-1)*100

cat(sprintf("%3d%11.4f%14.4f ( %4.4f%%)\n", num, round(average,4), round(analytical,4), round(error,2)))
}

Output:
N    average    analytical     (error)
===  =========  ============  ==========
1     1.0000        1.0000  ( 0.0000%)
2     1.5002        1.5000  ( 0.0100%)
3     1.8892        1.8889  ( 0.0100%)
4     2.2190        2.2188  ( 0.0100%)
5     2.5108        2.5104  ( 0.0200%)
6     2.7751        2.7747  ( 0.0200%)
7     3.0177        3.0181  ( 0.0100%)
8     3.2472        3.2450  ( 0.0700%)
9     3.4582        3.4583  ( 0.0000%)
10     3.6600        3.6602  ( 0.0100%)
11     3.8530        3.8524  ( 0.0200%)
12     4.0366        4.0361  ( 0.0100%)
13     4.2085        4.2123  ( 0.0900%)
14     4.3814        4.3820  ( 0.0100%)
15     4.5446        4.5458  ( 0.0300%)
16     4.7063        4.7043  ( 0.0400%)
17     4.8555        4.8579  ( 0.0500%)
18     5.0099        5.0071  ( 0.0600%)
19     5.1567        5.1522  ( 0.0900%)
20     5.2940        5.2936  ( 0.0100%)

## Racket

#lang racket
(require (only-in math factorial))

(define (analytical n)
(for/sum ([i (in-range 1 (add1 n))])
(/ (factorial n) (expt n i) (factorial (- n i)))))

(define (test n times)
(define (count-times seen times)
(define x (random n))
(if (memq x seen) times (count-times (cons x seen) (add1 times))))
(/ (for/fold ([count 0]) ([i times]) (count-times '() count))
times))

(define (test-table max-n times)
(displayln " n avg theory error\n------------------------")
(for ([i (in-range 1 (add1 max-n))])
(define average (test i times))
(define theory (analytical i))
(define difference (* (abs (sub1 (/ average theory))) 100))
(displayln (~a (~a i #:width 2 #:align 'right)
" " (real->decimal-string average 4)
" " (real->decimal-string theory 4)
" " (real->decimal-string difference 4)
"%"))))

(test-table 20 10000)

Output:
n avg    theory error
------------------------
1 1.0000 1.0000 0.0000%
2 1.5082 1.5000 0.5467%
3 1.8966 1.8889 0.4082%
4 2.2251 2.2188 0.2862%
5 2.5138 2.5104 0.1354%
6 2.7582 2.7747 0.5943%
7 3.0253 3.0181 0.2373%
8 3.2293 3.2450 0.4844%
9 3.4602 3.4583 0.0545%
10 3.6831 3.6602 0.6252%
11 3.8459 3.8524 0.1680%
12 4.0348 4.0361 0.0316%
13 4.1896 4.2123 0.5400%
14 4.3555 4.3820 0.6054%
15 4.5678 4.5458 0.4838%
16 4.6950 4.7043 0.1968%
17 4.8524 4.8579 0.1126%
18 5.0224 5.0071 0.3063%
19 5.1017 5.1522 0.9801%
20 5.3316 5.2936 0.7181%

## REXX

This REXX program automatically adjusts the precision (decimal digits) to be used based on the size of the
factorial (product) for   RUNS.

Also note that the   !   (factorial function)   uses memoization for optimization.

/*REXX program computes the average loop length mapping a random field 1···N ───► 1···N */
parse arg runs tests seed . /*obtain optional arguments from the CL*/
if runs =='' | runs =="," then runs = 40 /*Not specified? Then use the default.*/
if tests =='' | tests =="," then tests= 1000000 /* " " " " " " */
if datatype(seed, 'W') then call random ,, seed /*Is integer? For RAND repeatability.*/
!.=0;  !.0=1 /*used for factorial (!) memoization.*/
numeric digits 100000 /*be able to calculate 25k! if need be.*/
numeric digits max(9, length( !(runs) ) ) /*set the NUMERIC DIGITS for  !(runs). */
say right( runs, 24) 'runs' /*display number of runs we're using.*/
say right( tests, 24) 'tests' /* " " " tests " " */
say right( digits(), 24) 'digits' /* " " " digits " " */
say
say " N average exact  % error " /* ◄─── title, header ►────────┐ */
hdr=" ═══ ═════════ ═════════ ═════════"; pad=left('',3) /* ◄────────┘ */
say hdr
do #=1 for runs; av=fmtD( exact(#) ) /*use four digits past decimal point. */
xa=fmtD( exper(#) ) /* " " " " " " */
say right(#,9) pad xa pad av pad fmtD( abs(xa-av) * 100 / av) /*show values.*/
end /*#*/
say hdr /*display the final header (some bars).*/
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
!: procedure expose !.; parse arg z; if !.z\==0 then return !.z
!=1; do j=2 for z -1;  !=!*j;  !.j=!; end; /*compute factorial*/ return !
/*──────────────────────────────────────────────────────────────────────────────────────*/
exact: parse arg x; s=0; do j=1 for x; s=s + !(x) / !(x-j) / x**j; end; return s
/*──────────────────────────────────────────────────────────────────────────────────────*/
exper: parse arg n; k=0; do tests; \$.=0 /*do it TESTS times.*/
do n; r=random(1, n); if \$.r then leave
\$.r=1; k=k + 1 /*bump the counter. */
end /*n*/
end /*tests*/
return k/tests
/*──────────────────────────────────────────────────────────────────────────────────────*/
fmtD: parse arg y,d; d=word(d 4, 1); y=format(y, , d); parse var y w '.' f
if f=0 then return w || left('', d +1); return y
output   when using the default inputs:
40 runs
1000000 tests
48 digits

N    average     exact     % error
═══  ═════════  ═════════  ═════════
1     1          1          0
2     1.4964     1.5000     0.2400
3     1.8876     1.8889     0.0688
4     2.2222     2.2188     0.1532
5     2.5104     2.5104     0
6     2.7758     2.7747     0.0396
7     3.0194     3.0181     0.0431
8     3.2608     3.2450     0.4869
9     3.4565     3.4583     0.0520
10     3.6583     3.6602     0.0519
11     3.8513     3.8524     0.0286
12     4.0401     4.0361     0.0991
13     4.2133     4.2123     0.0237
14     4.3835     4.3820     0.0342
15     4.5445     4.5458     0.0286
16     4.6672     4.7043     0.7886
17     4.8575     4.8579     0.0082
18     5.0105     5.0071     0.0679
19     5.1517     5.1522     0.0097
20     5.2903     5.2936     0.0623
21     5.4328     5.4315     0.0239
22     5.5674     5.5662     0.0216
23     5.6990     5.6979     0.0193
24     5.8353     5.8268     0.1459
25     5.9536     5.9530     0.0101
26     6.0801     6.0767     0.0560
27     6.1997     6.1981     0.0258
28     6.3197     6.3173     0.0380
29     6.4328     6.4344     0.0249
30     6.5485     6.5495     0.0153
31     6.6615     6.6627     0.0180
32     6.7102     6.7740     0.9418
33     6.8826     6.8837     0.0160
34     6.9878     6.9917     0.0558
35     7.0996     7.0982     0.0197
36     7.2054     7.2032     0.0305
37     7.3073     7.3067     0.0082
38     7.4089     7.4088     0.0013
39     7.5052     7.5096     0.0586
40     7.6151     7.6091     0.0789
═══  ═════════  ═════════  ═════════

## Ruby

Ruby does not have a factorial method, not even in it's math library.

class Integer
def factorial
self == 0 ? 1 : (1..self).inject(:*)
end
end

def rand_until_rep(n)
rands = {}
loop do
r = rand(1..n)
return rands.size if rands[r]
rands[r] = true
end
end

runs = 1_000_000

puts " N average exp. diff ",
"=== ======== ======== ==========="
(1..20).each do |n|
sum_of_runs = runs.times.inject(0){|sum, _| sum += rand_until_rep(n)}
avg = sum_of_runs / runs.to_f
analytical = (1..n).inject(0){|sum, i| sum += (n.factorial / (n**i).to_f / (n-i).factorial)}
puts "%3d  %8.4f  %8.4f (%8.4f%%)" % [n, avg, analytical, (avg/analytical - 1)*100]
end
Output:
N    average    exp.        diff
===  ========  ========  ===========
1    1.0000    1.0000  (  0.0000%)
2    1.4999    1.5000  ( -0.0054%)
3    1.8886    1.8889  ( -0.0158%)
4    2.2181    2.2188  ( -0.0293%)
5    2.5107    2.5104  (  0.0110%)
6    2.7717    2.7747  ( -0.1074%)
7    3.0167    3.0181  ( -0.0484%)
8    3.2442    3.2450  ( -0.0257%)
9    3.4597    3.4583  (  0.0394%)
10    3.6572    3.6602  ( -0.0821%)
11    3.8502    3.8524  ( -0.0562%)
12    4.0357    4.0361  ( -0.0084%)
13    4.2139    4.2123  (  0.0360%)
14    4.3805    4.3820  ( -0.0360%)
15    4.5481    4.5458  (  0.0505%)
16    4.7030    4.7043  ( -0.0265%)
17    4.8582    4.8579  (  0.0075%)
18    5.0078    5.0071  (  0.0151%)
19    5.1568    5.1522  (  0.0893%)
20    5.2885    5.2936  ( -0.0961%)

## Rust

Library: rand
extern crate rand;

use rand::distributions::{IndependentSample, Range};
use std::collections::HashSet;
use std::env;
use std::process;

fn help() {
println!("usage: average_loop_length <max_N> <trials>");
}

fn main() {
let args: Vec<String> = env::args().collect();
let mut max_n: u32 = 20;
let mut trials: u32 = 1000;

match args.len() {
1 => {}
3 => {
max_n = args[1].parse::<u32>().unwrap();
trials = args[2].parse::<u32>().unwrap();
}
_ => {
help();
process::exit(0);
}
}

let mut rng = thread_rng();

println!(" N average analytical (error)");
println!("=== ========= ============ =========");
for n in 1..(max_n + 1) {
let the_analytical = analytical(n);
let the_empirical = empirical(n, trials, &mut rng);
println!(" {:>2} {:3.4} {:3.4} ( {:>+1.2}%)",
n,
the_empirical,
the_analytical,
100f64 * (the_empirical / the_analytical - 1f64));
}
}

fn factorial(n: u32) -> f64 {
(1..n + 1).fold(1f64, |p, n| p * n as f64)
}

fn analytical(n: u32) -> f64 {
let sum: f64 = (1..(n + 1))
.map(|i| factorial(n) / (n as f64).powi(i as i32) / factorial(n - i))
.fold(0f64, |a, v| a + v);
sum
}

fn empirical(n: u32, trials: u32, rng: &mut ThreadRng) -> f64 {
let sum: f64 = (0..trials)
.map(|_t| {
let mut item = 1u32;
let mut seen = HashSet::new();
let range = Range::new(1u32, n + 1);

for step in 0..n {
if seen.contains(&item) {
return step as f64;
}
seen.insert(item);
item = range.ind_sample(rng);
}
n as f64
})
.fold(0f64, |a, v| a + v);
sum / trials as f64
}

Output:

Using default arguments:

N    average    analytical    (error)
===  =========  ============  =========
1     1.0000        1.0000  ( +0.00%)
2     1.4992        1.5000  ( -0.05%)
3     1.8881        1.8889  ( -0.04%)
4     2.2177        2.2188  ( -0.05%)
5     2.5107        2.5104  ( +0.01%)
6     2.7752        2.7747  ( +0.02%)
7     3.0172        3.0181  ( -0.03%)
8     3.2452        3.2450  ( +0.01%)
9     3.4628        3.4583  ( +0.13%)
10     3.6606        3.6602  ( +0.01%)
11     3.8515        3.8524  ( -0.02%)
12     4.0348        4.0361  ( -0.03%)
13     4.2105        4.2123  ( -0.04%)
14     4.3835        4.3820  ( +0.03%)
15     4.5477        4.5458  ( +0.04%)
16     4.7042        4.7043  ( -0.00%)
17     4.8580        4.8579  ( +0.00%)
18     5.0076        5.0071  ( +0.01%)
19     5.1554        5.1522  ( +0.06%)
20     5.2911        5.2936  ( -0.05%)

## Scala

import scala.util.Random

object AverageLoopLength extends App {

val factorial: Stream[Double] = 1 #:: factorial.zip(Stream.from(1)).map(n => n._2 * factorial(n._2 - 1))

def expected(n: Int) = (for (i <- 1 to n) yield factorial(n) / Math.pow(n, i) / factorial(n - i)).sum

def trial(n: Int):Double = {
var count = 0
var x = 1
var bits = 0

while ((bits & x) == 0) {
count = count + 1
bits = bits | x
x = 1 << Random.nextInt(n)
}
count
}

def tested(n: Int, times: Int) = (for (i <- 1 to times) yield trial(n)).sum / times

val results = for (n <- 1 to 20;
avg = tested(n, 1000000);
theory = expected(n)
) yield (n, avg, theory, (avg / theory - 1) * 100)

println("n avg exp diff")
println("------------------------------------")
results foreach { n => {
println(f"\${n._1}%2d \${n._2}%2.6f \${n._3}%2.6f \${n._4}%2.3f%%")
}
}

}

Output:
n          avg         exp      diff
------------------------------------
1    1.000000    1.000000    0.000%
2    1.499894    1.500000    -0.007%
3    1.887826    1.888889    -0.056%
4    2.217514    2.218750    -0.056%
5    2.510049    2.510400    -0.014%
6    2.773658    2.774691    -0.037%
7    3.016585    3.018139    -0.051%
8    3.246865    3.245018    0.057%
9    3.458683    3.458316    0.011%
10    3.660361    3.660216    0.004%
11    3.852663    3.852372    0.008%
12    4.036970    4.036074    0.022%
13    4.213653    4.212348    0.031%
14    4.385226    4.382029    0.073%
15    4.545667    4.545807    -0.003%
16    4.705559    4.704258    0.028%
17    4.854056    4.857871    -0.079%
18    5.007146    5.007063    0.002%
19    5.148767    5.152196    -0.067%
20    5.292875    5.293585    -0.013%

## Scheme

(import (scheme base)
(scheme write)
(srfi 1 lists)
(only (srfi 13 strings) string-pad-right)
(srfi 27 random-bits))

(define (analytical-function n)
(define (factorial n)
(fold * 1 (iota n 1)))
;
(fold (lambda (i sum)
(+ sum
(/ (factorial n) (expt n i) (factorial (- n i)))))
0
(iota n 1)))

(define (simulation n runs)
(define (single-simulation)
(random-source-randomize! default-random-source)
(let ((vec (make-vector n #f)))
(let loop ((count 0)
(num (random-integer n)))
(if (vector-ref vec num)
count
(begin (vector-set! vec num #t)
(loop (+ 1 count)
(random-integer n)))))))
;;
(let loop ((total 0)
(run runs))
(if (zero? run)
(/ total runs)
(loop (+ total (single-simulation))
(- run 1)))))

(display " N average formula (error) \n")
(display "=== ========= ========= =========\n")
(for-each
(lambda (n)
(let ((simulation (inexact (simulation n 10000)))
(formula (inexact (analytical-function n))))
(display
(string-append
" "
(string-pad-right (number->string n) 3)
" "
(string-pad-right (number->string simulation) 6)
" "
(string-pad-right (number->string formula) 6)
" ("
(number->string (* 100 (/ (- simulation formula) formula)))
5)
"%)"))
(newline)))
(iota 20 1))

Output:
N   average   formula   (error)
=== ========= ========= =========
1     1.0      1.0      (0.0  %)
2     1.5018   1.5      (0.120%)
3     1.8863   1.8888   (-0.13%)
4     2.2154   2.2187   (-0.15%)
5     2.5082   2.5104   (-0.08%)
6     2.7613   2.7746   (-0.48%)
7     3.036    3.0181   (0.591%)
8     3.2656   3.2450   (0.634%)
9     3.455    3.4583   (-0.09%)
10    3.682    3.6602   (0.595%)
11    3.8233   3.8523   (-0.75%)
12    4.0409   4.0360   (0.119%)
13    4.2471   4.2123   (0.825%)
14    4.3577   4.3820   (-0.55%)
15    4.5351   4.5458   (-0.23%)
16    4.7181   4.7042   (0.294%)
17    4.8877   4.8578   (0.614%)
18    5.0239   5.0070   (0.336%)
19    5.1216   5.1521   (-0.59%)
20    5.2717   5.2935   (-0.41%)

## Simula

BEGIN

REAL PROCEDURE FACTORIAL(N); INTEGER N;
BEGIN
REAL RESULT;
INTEGER I;
RESULT := 1.0;
FOR I := 2 STEP 1 UNTIL N DO
RESULT := RESULT * I;
FACTORIAL := RESULT;
END FACTORIAL;

REAL PROCEDURE ANALYTICAL (N); INTEGER N;
BEGIN
REAL SUM, RN;
INTEGER I;
RN := N;
FOR I := 1 STEP 1 UNTIL N DO
BEGIN
SUM := SUM + FACTORIAL(N) / FACTORIAL(N - I) / RN ** I;
END;
ANALYTICAL := SUM;
END ANALYTICAL;

REAL PROCEDURE EXPERIMENTAL(N); INTEGER N;
BEGIN
INTEGER NUM;
INTEGER COUNT;
INTEGER RUN;
FOR RUN := 1 STEP 1 UNTIL TESTS DO
BEGIN
BOOLEAN ARRAY BITS(1:N);
INTEGER I;
FOR I := 1 STEP 1 UNTIL N DO
BEGIN
NUM := RANDINT(1,N,SEED);
IF BITS(NUM) THEN GOTO L;
BITS(NUM) := TRUE;
COUNT := COUNT + 1;
END FOR I;
L:
END FOR RUN;
EXPERIMENTAL := COUNT / TESTS;
END EXPERIMENTAL;

INTEGER SEED, TESTS;
SEED := ININT;
TESTS := 1000000;
BEGIN
REAL A, E, ERR;
INTEGER I;
OUTTEXT(" N AVG CALC  %DIFF"); OUTIMAGE;
FOR I := 1 STEP 1 UNTIL 20 DO
BEGIN
A := ANALYTICAL(I);
E := EXPERIMENTAL(I);
ERR := (ABS(E-A)/A)*100.0;
OUTINT(I, 2);
OUTFIX(E, 4, 7);
OUTFIX(A, 4, 10);
OUTFIX(ERR, 4, 10);
OUTIMAGE;
END FOR I;
END;
END
Input:
678
Output:
N  AVG    CALC   %DIFF
1 1.0000    1.0000    0.0000
2 1.4999    1.5000    0.0075
3 1.8890    1.8889    0.0072
4 2.2182    2.2188    0.0243
5 2.5105    2.5104    0.0027
6 2.7746    2.7747    0.0025
7 3.0164    3.0181    0.0590
8 3.2447    3.2450    0.0110
9 3.4567    3.4583    0.0453
10 3.6622    3.6602    0.0539
11 3.8503    3.8524    0.0546
12 4.0373    4.0361    0.0300
13 4.2105    4.2123    0.0445
14 4.3819    4.3820    0.0027
15 4.5475    4.5458    0.0376
16 4.7056    4.7043    0.0295
17 4.8559    4.8579    0.0396
18 5.0105    5.0071    0.0694
19 5.1541    5.1522    0.0376
20 5.2961    5.2936    0.0467

## Seed7

\$ include "seed7_05.s7i";
include "float.s7i";

const integer: TESTS is 1000000;

const func float: factorial (in integer: number) is func
result
var float: factorial is 1.0;
local
var integer: i is 0;
begin
for i range 2 to number do
factorial *:= flt(i);
end for;
end func;

const func float: analytical (in integer: number) is func
result
var float: sum is 0.0;
local
var integer: i is 0;
begin
for i range 1 to number do
sum +:= factorial(number) / factorial(number - i) / flt(number)**i;
end for;
end func;

const func float: experimental (in integer: number) is func
result
var float: experimental is 0.0;
local
var integer: run is 0;
var set of integer: seen is EMPTY_SET;
var integer: current is 1;
var integer: count is 0;
begin
for run range 1 to TESTS do
current := 1;
seen := EMPTY_SET;
while current not in seen do
incr(count);
incl(seen, current);
current := rand(1, number);
end while;
end for;
experimental := flt(count) / flt(TESTS);
end func;

const proc: main is func
local
var integer: number is 0;
var float: analytical is 0.0;
var float: experimental is 0.0;
var float: err is 0.0;
begin
writeln(" N avg calc  %diff");
for number range 1 to 20 do
analytical := analytical(number);
experimental := experimental(number);
err := abs(experimental - analytical) / analytical * 100.0;
writeln(number lpad 2 <& experimental digits 4 lpad 7 <&
analytical digits 4 lpad 7 <& err digits 3 lpad 7);
end for;
end func;
Output:
N  avg    calc   %diff
1 1.0000 1.0000  0.000
2 1.4999 1.5000  0.005
3 1.8891 1.8889  0.010
4 2.2196 2.2188  0.040
5 2.5073 2.5104  0.122
6 2.7744 2.7747  0.010
7 3.0186 3.0181  0.015
8 3.2463 3.2450  0.040
9 3.4592 3.4583  0.027
10 3.6597 3.6602  0.013
11 3.8549 3.8524  0.066
12 4.0374 4.0361  0.033
13 4.2115 4.2123  0.019
14 4.3835 4.3820  0.033
15 4.5474 4.5458  0.035
16 4.7017 4.7043  0.055
17 4.8558 4.8579  0.043
18 5.0096 5.0071  0.051
19 5.1522 5.1522  0.000
20 5.2907 5.2936  0.054

## Tcl

# Generate a list of the numbers increasing from \$a to \$b
proc range {a b} {
for {set result {}} {\$a <= \$b} {incr a} {lappend result \$a}
return \$result
}

# Computing the expected value analytically
proc tcl::mathfunc::factorial n {
::tcl::mathop::* {*}[range 2 \$n]
}
proc Analytical {n} {
set sum 0.0
foreach x [range 1 \$n] {
set sum [expr {\$sum + factorial(\$n) / factorial(\$n-\$x) / double(\$n)**\$x}]
}
return \$sum
}

# Determining an approximation to the value experimentally
proc Experimental {n numTests} {
set count 0
set u0 [lrepeat \$n 1]
foreach run [range 1 \$numTests] {
set unseen \$u0
for {set i 0} {[lindex \$unseen \$i]} {incr count} {
lset unseen \$i 0
set i [expr {int(rand()*\$n)}]
}
}
return [expr {\$count / double(\$numTests)}]
}

# Tabulate the results in exactly the original format
puts " N average analytical (error)"
puts "=== ========= ============ ========="
foreach n [range 1 20] {
set a [Analytical \$n]
set e [Experimental \$n 100000]
puts [format "%3d  %9.4f  %12.4f (%6.2f%%)" \$n \$e \$a [expr {abs(\$e-\$a)/\$a*100.0}]]
}
Output:
N    average    analytical    (error)
===  =========  ============  =========
1     1.0000        1.0000  (  0.00%)
2     1.5003        1.5000  (  0.02%)
3     1.8881        1.8889  (  0.04%)
4     2.2228        2.2188  (  0.18%)
5     2.5109        2.5104  (  0.02%)
6     2.7804        2.7747  (  0.20%)
7     3.0223        3.0181  (  0.14%)
8     3.2456        3.2450  (  0.02%)
9     3.4598        3.4583  (  0.04%)
10     3.6590        3.6602  (  0.03%)
11     3.8527        3.8524  (  0.01%)
12     4.0390        4.0361  (  0.07%)
13     4.2156        4.2123  (  0.08%)
14     4.3821        4.3820  (  0.00%)
15     4.5527        4.5458  (  0.15%)
16     4.6952        4.7043  (  0.19%)
17     4.8530        4.8579  (  0.10%)
18     4.9912        5.0071  (  0.32%)
19     5.1578        5.1522  (  0.11%)
20     5.2992        5.2936  (  0.11%)

## Unicon

Translation of: C

\$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
Output:
n      avg     exp.    diff
-----------------------------
1   1.0000   1.0000  0.000%
2   1.5008   1.5000  0.056%
3   1.8879   1.8889 -0.051%
4   2.2208   2.2188  0.091%
5   2.5127   2.5104  0.093%
6   2.7759   2.7747  0.044%
7   3.0175   3.0181 -0.023%
8   3.2425   3.2450 -0.079%
9   3.4571   3.4583 -0.034%
10   3.6613   3.6602  0.029%
11   3.8493   3.8524 -0.081%
12   4.0384   4.0361  0.058%
13   4.2133   4.2123  0.023%
14   4.3804   4.3820 -0.037%
15   4.5475   4.5458  0.038%
16   4.7049   4.7043  0.014%
17   4.8575   4.8579 -0.008%
18   5.0088   5.0071  0.035%
19   5.1533   5.1522  0.021%
20   5.2893   5.2936 -0.081%

## VBA

Translation of: Phix
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
Output:
n     avg.     exp.  (error%)
==   ======   ======  ========
1  1,0000   1,0000  (0,000%)
2  1,4994   1,5000  (0,041%)
3  1,8893   1,8889  (0,023%)
4  2,2187   2,2188  (0,001%)
5  2,5107   2,5104  (0,010%)
6  2,7769   2,7747  (0,080%)
7  3,0162   3,0181  (0,064%)
8  3,2472   3,2450  (0,066%)
9  3,4603   3,4583  (0,056%)
10  3,6577   3,6602  (0,070%)
11  3,8527   3,8524  (0,010%)
12  4,0361   4,0361  (0,001%)
13  4,2121   4,2123  (0,005%)
14  4,3825   4,3820  (0,010%)
15  4,5466   4,5458  (0,016%)
16  4,7023   4,7043  (0,041%)
17  4,8567   4,8579  (0,025%)
18  5,0031   5,0071  (0,079%)
19  5,1530   5,1522  (0,016%)
20  5,2958   5,2936  (0,041%)

## zkl

const N=20;

(" N average analytical (error)").println();
("=== ========= ============ =========").println();
foreach n in ([1..N]){
a := avg(n);
b := ana(n);
"%3d  %9.4f  %12.4f (%6.2f%%)".fmt(
n, a, b, ((a-b)/b*100)).println();
}

fcn f(n){ (0).random(n) }

fcn avg(n){
tests := 0d10_000;
sum := 0;
do(tests){
v:=(0).pump(n,List,T(Void,False)).copy();
while(1){
z := f(n);
if(v[z]) break;
v[z] = True;
sum += 1;
}
}
return(sum.toFloat() / tests);
}

fcn fact(n) { (1).reduce(n,fcn(N,n){N*n},1.0) } //-->Float
fcn ana(n){
n=n.toFloat();
(1).reduce(n,'wrap(sum,i){ sum+fact(n)/n.pow(i)/fact(n-i) },0.0);
}
Output:
N    average    analytical    (error)
===  =========  ============  =========
1     1.0000        1.0000  (  0.00%)
2     1.5053        1.5000  (  0.35%)
3     1.8899        1.8889  (  0.05%)
4     2.2384        2.2188  (  0.89%)
5     2.5090        2.5104  ( -0.06%)
6     2.7824        2.7747  (  0.28%)
7     3.0449        3.0181  (  0.89%)
8     3.2430        3.2450  ( -0.06%)
9     3.4744        3.4583  (  0.47%)
10     3.6693        3.6602  (  0.25%)
11     3.8833        3.8524  (  0.80%)
12     4.0225        4.0361  ( -0.34%)
13     4.1899        4.2123  ( -0.53%)
14     4.4135        4.3820  (  0.72%)
15     4.5807        4.5458  (  0.77%)
16     4.7304        4.7043  (  0.56%)
17     4.8437        4.8579  ( -0.29%)
18     4.9838        5.0071  ( -0.46%)
19     5.1767        5.1522  (  0.48%)
20     5.2723        5.2936  ( -0.40%)