Permutations, Adjacents Have Near-by Binary 1's

[Leroy Quet]

First, I want to thank Klaus Brockhaus for calculating and submitting two 
recent sequences of mine. (And I thank anyone who has submitted, 
extended, or commented on any of my sequences.)


I have another idea for a sequence, and I hope someone can extend it and 
submit it if it is not already in the EIS.

Take the integers 1 through n written in binary.
The nth term of the sequence is the number of permutations of these 
binary numbers such that every 1-digit in the kth element (2<=k<=n) of 
the permutation is either in the same position as, is one position to the 
left of, or is one position to the right of a 1-digit in the (k-1)th 
element written in binary.

So, for example, 4 = 100 (in binary) can follow 3 = 011 (in binary), 
because the 1-digit in 4 is one position to the left of the leftmost 
1-digit in 3.

But, on the other hand, 3 cannot follow 4, because the rightmost 1-digit 
in 3 is two positions from the 1-digit in 4.

So, an allowable permutation for n=4 is:

2,1,3,4 =
010
001
011
100 in binary.

But 

2,3,4,1 = 
010
011
100
001 in binary,

is not allowable.

I get that the sequence of number of such permutations for 1 through n
begins 1, 2, 6, 8,...

thanks,
Leroy Quet