Transform involving permutations

[David Wasserman]

The "Quet transform" of A006519 is
1,2,1,2,3,2,1,2,3,2,3,4,3,2,1,2,3,2,3,4,3,2,3,4,3,4,5,4,3,2,1...
which is constructed as follows: 
Start with a subsequence consisting of a single 1, and repeatedly apply the operation S -> S,1+S,1
thus
1
1,2,1
1,2,1,2,3,2,1
1,2,1,2,3,2,1,2,3,2,3,4,3,2,1
etc.
This appears to be 1 + A080468.

%I A006519 M0162
%S A006519 1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,16,1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,32,1,2,
%T A006519 1,4,1,2,1,8,1,2,1,4,1,2,1,16,1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,64,1,2,1,4,
%U A006519 1,2,1,8,1,2,1,4,1,2,1,16,1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,32,1,2,1,4,1,2
%N A006519 Highest power of 2 dividing n.

%I A080468
%S A080468 0,1,0,1,2,1,0,1,2,1,2,3,2,1,0,1,2,1,2,3,2,1,2,3,2,3,4,3,2,1,0,1,2,1,2,
%T A080468 3,2,1,2,3,2,3,4,3,2,1,2,3,2,3,4,3,2,3,4,3,4,5,4,3,2,1,0,1,2,1,2,3,2,1,
%U A080468 2,3,2,3,4,3,2,1,2,3,2,3,4,3,2,3,4,3,4,5,4,3,2,1,2,3,2,3,4,3,2,3,4,3,4
%N A080468 a(n) = A080578(n)-2n.
%C A080468 Terms between 2^n and 2^(n+1) as n goes to infinity tend to the sequence : 0,1,2,1,2,3,2,1,2,3,2,3,4,3,2,1,2\
  ,3,2,3,4,3,2,3,4,3,4,5,4,3,2,1,....