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I need help with the following problem :

In how many ways we can seat $n$ couples on the two longer sides of a rectangular table such that no couple sits next to each other and facing each other, and people of the same gender do not sit next to each other or facing each other.

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The "non-sexist solution of the ménage problem" by Bogart and Doyle gives a straightforward description of how to reduce a related problem to one of counting arrangements of dominoes. (Your exact problem, in fact, is one of their exercises for the reader.) – mjqxxxx Oct 28 '11 at 6:50
I'd like to up-vote the edit ... the original invited an unfortunate alternative understanding. – Mark Bennet Oct 28 '11 at 8:32
I guess we're assuming no same-gender couples? – Gerry Myerson Oct 28 '11 at 9:58

One approach is to find the answer by hand for $n=4,5,6$, say, then look it up in the Online Encyclopedia of Integer Sequences.

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