No label defined (Q6645084)

For a finite set \(A\) of positive integers and a given positive integer \(k\), define \(\omega_k(A) = \sum_{i\in A, i\neq k}1\). For each positive integer \(n\), consider\N\[\Na_{k, n} = |\{E \subset\mathbb{N}: E = \emptyset\; \text{ or }\; \omega_k(E)< \min E \leq \max E \leq n\}|.\N\]\NIn this paper, the authors show that\N\[\Na_{k, k+\ell} = 2F_{k + \ell}\, ,\text{ for all }\ell \geq 0\; \text{ and } \; k \geq \ell + 2\, ,\N\]\Nwhere \(F_n\) is the \(n\)th Fibonacci number. The paper generalizes several related results namely those by \textit{K. Beanland} et al. [Fibonacci Q. 60, No. 4, 352--356 (2022; Zbl 1512.11020)].