# All my bags are packed, I'm ready to go

A subset $$S$$ of $$\{1,2,\ldots,n\}$$ is said to be packed if whenever $$i, j \in S$$ the number $$\left\lfloor \frac{i+j}{2} \right\rfloor$$ is also in $$S.$$ Determine how many subsets of $$\{1,2,\ldots, 25\}$$ are packed.

Details and assumptions

$$i$$ and $$j$$ need not be distinct. If $$i= j$$ is in the set, then clearly so is $$\left\lfloor \frac{i+j}{2} \right\rfloor$$.

The sets $$S$$ and the empty set clearly satisfy the conditions of the question, and should be included in your count.

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