$B_i$ is a subset of $\mathbb{N}$ such that
- $$\sum_{x \in B_i}x = i$$
- $$ \text{card}(B_i) \leq i$$
- $$\forall x \in B_i \:\: x\leq i $$
$i$ is a non-zero integer
my goal is to find a general formula for the number of possible subsets that satisfy those 3 conditions.
example : if $i = 3 $ then the number of solutions is $3$ because $1+1+1 = 1+2 = 3$
if $i = 4 $ then the number of solutions is $5$ because $1+1+1+1 = 1+1+2=1+3=2+2 = 4$
I tried coding an algorithm to test all the possible cases and thereupon noticing a pattern or something but if $i$ is greater than $5$ it just takes too much time... it never ends.
any help would be greately appreciated. thanks.