How many ways can $n$ numbers from 1 to $n$ be arranged in a circular order without consecutive integers being placed together? (Note: 1 and $n$ are also considered consecutive integers)
For example, if $n=5$, there are 2 ways: 53142 and 24135 (in a circle); and if $n=6$, I get 6 ways: 531462 and 264135, 241536 and 635142, 631524 and 425136 (in a circle).
I have tried inclusion-exclusion principle but I have found it's difficult to get the expression; and have also failed to get its recursive equation. Any help in this matter will be appreciated!