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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