Consider a convex \((n+2)\)-gon. Let \(a_n\) denote the number of ways to add non-crossing chords to this polygon, including the case where no chords are added (i.e., \(a_0=a_1=0\) and \(a_2=3\)).
Find a recurrence relation for the sequence \(a_n\) and determine its generating function.
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