Determine the number of circular permutations of the set $\{1,2\dots n\}$ with no two consecutive integers appearing adjacently

Question

The question is: Determine the number of ways of placing the numbers $1, 2 \dots n$ in a circle such that no two consecutive integers ($1$ and $n$ are considered consecutive) appear adjacently.

The answer (determined by an R code) is given by the OEIS sequence A078603 and has no known recurrence on the website. Can someone help me determine a closed form/determine a recurrence for the given problem if it exists and is known?

Answer 1

I can give you an explicit answer, but not a recurrence. For $n\ge 3$, the number of ways is $$ {(n-1)!}+2\cdot (-1)^n+\sum_{k=1}^{n-1}(-1)^k \sum_{t=1}^k\frac{n}t\binom{k-1}{t-1}\binom{n-k-1}{t-1}2^{t}(n-k-1)! $$ I gave a proof of this in an earlier answer. Here is a sanity check using Python: Try it online!