We consider the lattice of order ideals of the union of an \(n\)-element fence and an antichain of size \(i\), whose Hasse diagram turns out to be isomorphic to the \(i\)-th extended Fibonacci cube. We prove that the Whitney numbers of these lattices form a unimodal sequence satisfying a particular property, called \({alternating}\), we find the maximum level of the game sequence and determine the exact values of these numbers.