Permutations: Adjacent Elements Are Coprime

[Leroy Quet]

I wrote to seq.fan about this a while back, perhaps, but I cannot 
remember what kind of replies, if any, I received.

Could someone calculate/submit the sequence, if it is not already in the 
database, where the nth term is the number of permutations of 
(1,2,3,...,n) where each integer is coprime with its closest neighbors in 
the permutation.


I get, by hand (so I might have made a mistake), the sequence beginning:
1,2,6,12,72,68,...

For example,

for n = 6, we can have in the count the permutation
5,4,3,2,1,6,

but not the permutation
2,5,4,1,6,3
(because here 3 is adjacent with 6).

thanks,
Leroy Quet