Weighted Schreier-type Sets and the Fibonacci Sequence

For a finite set A and k∈ N, let ωₖ(A) = ∑i∈ A, i≠ k1. For each n∈ N, define ak, n\ =\ |⊂ N\,:\, E = ∅ or ωₖ(E) < min E max E n\|. First, we prove that ak,k+ \ =\ 2Fₖ₊, for all 0 and k +2, where Fₙ is the nth Fibonacci number. Second, we show that |⊂ N\,:\, max E = n+1, min E > ω2,3(E), and |E|≠ 2\|\ =\ Fₙ, where ω2,3(E) = ∑i∈ E, i≠ 2, 31.