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Easy count (Posted on 2014-07-29) Difficulty: 3 of 5
Let S be a set of some positive integers.
We'll call S autonomous if the number of elements in S is itself an element of S. e.g. the set {2,3,5} is autonomous, as is the set {2,7}, but the sets {1, 4} or {2,4,5} are not.

Determine a general formula for the number
of autonomous subsets of {1, 2, 3, ... , n}.

See The Solution Submitted by Ady TZIDON    
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Solution computer assisted solution | Comment 1 of 5
Let f(n) be the sought function of n.

f(1) = 0
f(2) = 1  {2}
f(3) = 3  {2}{3}{1,3}
f(4) = 7  {2}{3}{4}{1,3}{1,4}{3,4}{1,2,4}

f(n) = Sigma{i=1 to n-1} C(n-1,i)

   10   for N=2 to 12
   15     T=0
   20     for I=1 to N-1
   30        T=T+combi(N-1,I)
   40     next
   50     print N,T
   60   next

finds 

2    1
3    3
4    7
5    15
6    31
7    63
8    127
9    255
10   511
11   1023
12   2047

f(n) = 2^(n-1) - 1


  Posted by Charlie on 2014-07-29 14:38:40
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