You start a rumor by telling it to n people and each of them tells it to another n-1 people. If each of these people, in turn, tells the rumor to another n-2 people, and so forth with no person hearing the rumor from more than one source, find a closed-form expression (in terms of n) for the total number of people that have been told the rumor.

The following program keeps track of how many people are told in each generation of tellees (gen = previous generation size times the reduced number from n), and the overall number of people who have been told (told).

CLS
DEFDBL A-Z
FOR n = 1 TO 17
 gen = n: told = n
 FOR n1 = n - 1 TO 1 STEP -1
   gen = gen * n1
   told = told + gen
 NEXT
 PRINT USING "## ############### "; n; told
NEXT

The resulting table is:

 1               1
 2               4
 3              15
 4              64
 5             325
 6            1956
 7           13699
 8          109600
 9          986409
10         9864100
11       108505111
12      1302061344
13     16926797485
14    236975164804
15   3554627472075
16  56874039553216
17 966858672404689

As a manual check for, say 4:

You tell 4 people                           total 4 so far
4 people tell 3 people each, making 12           16
12 people tell 2 people each, making 24          40
24 people tell 1 person each, making 24          64 

Looking up the sequence in Sloane finds Sloane A007526, where a simple formula is given:

[e*n! - 1] where the square brackets indicate the floor function, or integer part: the greatest integer not exceeding the argument enclosed.

The following program calculates that formula and places it in a column to the right, for comparison.

CLS
DEFDBL A-Z
e = EXP(1)
nf = 1
FOR n = 1 TO 17
 nf = nf * n
 gen = n: told = n
 FOR n1 = n - 1 TO 1 STEP -1
   gen = gen * n1
   told = told + gen
 NEXT
 PRINT USING "## ############### ###############"; n; told; INT(e * nf - 1)
NEXT

 1               1               1
 2               4               4
 3              15              15
 4              64              64
 5             325             325
 6            1956            1956
 7           13699           13699
 8          109600          109600
 9          986409          986409
10         9864100         9864100
11       108505111       108505111
12      1302061344      1302061344
13     16926797485     16926797485
14    236975164804    236975164804
15   3554627472075   3554627472075
16  56874039553216  56874039553216
17 966858672404689 966858672404689

The columns do match, verifying the closed-form expression.

Posted by Charlie on 2008-04-25 16:09:17