Weighted Schreier-type Sets and the Fibonacci Sequence

For a finite set A and k N, let ₖ (A) = ₈ ₀, ₈ ₊1. For each n N, define a₊, ₍\ =\ |\E N\,: \, E = or ₖ (E) ₂, ₃ (E), and |E| 2\|\ =\ F₍, where ₂, ₃ (E) = ₈ ₄, ₈ ₂, ₃1.