TITLE: What's the n-th term of this sequence (partition numbers)?
QUESTION [6 upvotes]: For the sequence of unique partitions of integer $n$:
$n, \\
(n-1)+1,\\
(n-2)+2,\ (n-2)+1+1,\\
(n-3)+3,\ (n-3)+2+1,(n-3)+1+1+1,\\
(n-4)+4,\ (n-4)+3+1,\ (n-4)+2+2,\ (n-4)+2+1+1,\ (n-4)+1+1+1+1,\\
...etc.$
the sequence of the number of items in each partition is:
$1,2,2,3,2,3,4,2,3,3,4,5,2,3,3,4,4,5,6,2,3,3,4,3,4,....etc.$
What's the $nth$ term of this sequence?  Does a closed form expression for it exist?  My guess is not, but I can't prove it.  The need for one arises from the generating function for a permutation of a set of objects in a set of containers.

REPLY [2 votes]: Always try OEIS (Online Encyclopedia of Integer Sequences).  If you subtract one from your sequence you get 
$0, 1, 1, 2, 1, 2, 3, 1, 2, 2, 3, 4, \ldots$
This is sequence A036043 in OEIS. "Irregular triangle read by rows: row n (n >= 0) gives number of parts in all partitions of n (in Abramowitz and Stegun order)."  I doubt there is any kind of nice formula.
The Abramowitz and Stegun order  on partitions is apparently the one you are using.