Disjointness of proposed even/odd formulas for the OEIS A248982 Fibonacci-average sequence

Show that the intersection of the two sets S_even and S_odd is empty, where S_even := { n·F(n+3) − (n − 1)·F(n+2) : n ≥ 1 and n is even } and S_odd := { F(n+1+2) : n > 1 and n is odd }, with F(t) denoting the t-th Fibonacci number. Establishing this disjointness would validate the distinctness condition for the sequence of distinct least positive integers whose first-n-term average is a Fibonacci number, as proposed in the refinement of OEIS A248982.

Background

Section 2.9 discusses a greedily defined integer sequence and relates it to OEIS A248982, which is the sequence of distinct least positive integers such that the average of the first n terms is a Fibonacci number. The authors outline candidate closed-form expressions for the terms a_n when n is even and when n is odd.

To confirm the correctness of these proposed formulas for A248982, it is necessary to verify that the values they produce for even indices and for odd indices do not overlap; otherwise, the distinctness condition of the sequence could fail. The authors note that, although the proof strategy should resemble that used for the power-of-two average sequence in Theorem 12, they were unable to establish this crucial disjointness property.