Find the general term for the sequence of no. of permutations of derangements of length n where adjacent to an element it was originally adjacent to.

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I wonder if there are any means to obtain the general term for the sequence of no. of permutations of derangements of length n where adjacent to an element it was originally adjacent to. The sequence is A288208.

A related question

I have tried to use Graph Theory to deduce the relationship between A002464 and A288208, but I failed.

Is there any hints or ways to find out the general term?