Weighted Schreier-type Sets and the Fibonacci Sequence

For a finite set $A\subset \mathbb{N}$ and $k\in \mathbb{N}$, let ${\omega }_k(A)={\sum }_{i\in A,i\mathrm{\ne }k}1$. For each $n\in \mathbb{N}$, define $$a_{k,n}=\mathrm{\mid }\{E\subset \mathbb{N}:E=\mathrm{\varnothing }or{\omega }_k(E)<\min E⩽\max E⩽n\}\mathrm{\mid }\mathrm{.}$$ First, we prove that $$a_{k,k+\mathrm{\ell }}=2F_{k+\mathrm{\ell }},forall\mathrm{\ell }⩾0andk⩾\mathrm{\ell }+2,$$ where $F_n$ is the $n$th Fibonacci number. Second, we show that $$\mathrm{\mid }\{E\subset \mathbb{N}:\max E=n+1,\min E>{\omega }_{2,3}(E),and\mathrm{\mid }E\mathrm{\mid }\mathrm{\ne }2\}\mathrm{\mid }=F_n,$$ where ${\omega }_{2,3}(E)={\sum }_{i\in E,i\mathrm{\ne }2,3}1$.