Fat Subsets

Discrete Mathematics Level pending

A finite set of positive integers is called fat if each of its members is at least as large as the number of elements in the set. (The empty set is considered to be \(fat\).) Let \(a_n\) denote the number of fat subsets of \( \{1,2,\ldots,n \}\) that contain no two consecutive integers, and let \(b_n\) denote the number of subsets of \( \{1,2,\ldots,n \}\) in which any two elements differ by at least three.

Find \(a_{1729}-b_{1729}+1\).

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