Let \(\pi \) be a cyclic permutation that can be expressed in its one-line form as \(\pi = \pi _1\pi _2 \cdots \pi _n\) and in its cycle form as \(\pi = (c_1,c_2,\ldots , c_n)\) . Archer et al. introduced the notion of pattern avoidance of one-line and all cycle forms for a cyclic permutation \(\pi \) , defined as both \(\pi _1\pi _2 \cdots \pi _n\) and its arbitrary cycle form \(c_ic_{i+1}\cdots c_nc_1c_2\cdots c_{i-1}\) avoiding a given pattern. Let \(\mathcal {A}^\circ _n(\sigma ; \tau )\) denote the set of cyclic permutations in the symmetric group \(S_n\) that avoid \(\sigma \) in their one-line forms and avoid \(\tau \) in all their cycle forms. In this note, we prove that \(|\mathcal {A}^\circ _n(2431; 1324)|\) is the \((n-1)^\mathrm{{st}}\) Pell number for any positive integer n. Thereby, we give a positive answer to a conjecture of Archer et al.
