Number 3 can be expressed as the sum of one or more positive integers in 4 distinct ways:
3; 2 + 1; 1 + 2; 1 + 1 + 1
Number 4 can be expressed as the sum of one or more positive integers in 8 distinct ways:
4; 3 + 1; 1+3; 2 + 2; 2 + 1 + 1; 1+2+1; 1+1+2; 1+1+1+1
Prove : any positive integer n can be so expressed in 2^{n  1 }ways.
The number doubles when n is increased by 1. This is because there are two ways to make a sum of n become a sum of n+1:
(A) Add 1 at the end
(B) Add 1 to the last number
Using the given sums to 4, applying (1) to each gives the first 8
4+1; 3+1+1; 1+3+1; 2+2+1; 2+1+1+1; 1+2+1+1; 1+1+2+1; 1+1+1+1+1
Applying rule 2 to each gives 8 more
5; 3+2; 1+4; 2+3; 2+1+2; 1+2+2; 1+1+3; 1+1+1+2
The rules can't cause any repeats because with rule (A) they all end in 1 and with rule (B) they can't end in 1.
To finish this recursive rule, note that 1 can be done in 2^{0}=1 way: 1

Posted by Jer
on 20170326 14:33:59 