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

Probability Level 4

A subset SS of {1,2,,n}\{1,2,\ldots,n\} is said to be packed if whenever i,jSi, j \in S the number i+j2\left\lfloor \frac{i+j}{2} \right\rfloor is also in S.S. Determine how many subsets of {1,2,,25}\{1,2,\ldots, 25\} are packed.

Details and assumptions

ii and jj need not be distinct. If i=ji= j is in the set, then clearly so is i+j2 \left\lfloor \frac{i+j}{2} \right\rfloor.

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

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