How many tuples of numbers from [1..n] have the sum of its elements equal to n? [1..n] is the set of integers from 1 to n. The tuples can be of any finite length. The length of each tuple should range from 1 to n. I am asking how many tuples have elements such that the total sum of the elements is n. For example let's say the set is {1,2,3}. The number of tuples is 4: (1,2), (2,1), (1,1,1), (3).
(Thanks to André Nicolas, for pointing out a bug in an earlier version of this question.)
 A: The number of ways is $2^{n-1}$. We can show this by induction.
The sequences of numbers chosen from $1,2,\dots,n+1$ that have sum $n+1$ can be divided into two types:
A Type 1 sequence has last entry equal to $1$. It can be obtained from a unique  sequence $S$ with sum $n$ by appending  a $1$ to $S$. 
A Type $2$ sequence has last entry $\ge 1$. It can be obtained from a unique sequence $S$ with sum $n$ by adding $1$ to the last entry in the sequence $S$.   
Thus if $a_{n}$ is the number of sequences for $n$, then $a_{n+1}=2a_n$. 
Remark: A sequence of positive integers whose sum is $n$ is called a Composition of $n$. For much more detail, please see this Wikipedia article.
Your list for $3$ was incomplete, for $3$ all by itself is a sequence with sum $3$. If you do not wish to allow sequences of length $1$, the number you are looking for is $2^{n-1}-1$. 
A: Another way to see that the total number of tuples is $2^{n-1}$ is to imagine $n$ items in a row.  You may then insert or not insert a divider between any two of the items (there are $n-1$ potential insertion points, each with a choice of using or not using a divider).
For example, if $n=5$, you could have $**|*|*|*$ corresponding to the tuple (2, 1, 1, 1).
