A (New?) Sequence Transform

[Leroy Quet]

Actually, I forgot to ask (my main question) if there is a more DIRECT 
way to calculate the transform (of {a(k)} below into {b(k)})? 
  
 
It seems intuitive to me that the transform can be much more easily 
described.
 
     
Leroy
 
qqquet at mindspring.com (Leroy Quet) wrote in message 
news:<b4be2fdf.0312192008.98534e3 at posting.google.com>...
> Let {a(k)} be an infinite sequence of positive integers, and where {a}
> contains an infinite numbers of terms equal to 1.
> 
> Here is a simple(well...) transform which converts {a} into another
> infinite sequence {b(k)} of positive integers,
> and where {b} also has an infinite number of 1's.
> 
> 
> Let {c(k)} be a permutation of the positive integers, where
> c(1) = 1, and
> 
> c(m+1) = the a(m)_th yet-unpicked positive integer.
> (By "yet-unpicked",
> I mean that c(m) is not among c(1),c(2),c(3),...c(m-1).)
> 
> Let {d(m)} be the inverse of {c}.
> (ie. c(d(m)) = m for all m.)
> 
> 
> Apply the reverse of the {a}->{c} step to {d} to get {b}.
> In other words, 
> b(m) = the order among + integers 
> not in {d(1),d(2),d(3),...d(m)}
> of d(m+1).
> 
> So, this simpler-than-it-must-appear-to-you transform seems like it
> must have some uses, but I do not know.
> 
> An example of it being used:
> 
> Sequence A001222 of EIS:
> 
> (from a(2) on)
> 
> a ->  1,1,2,1,2,1,3,2,2,1,3,...
> c -> 1,2,3,5,4,7,6,10,9,11,8,14,...
> d -> 1,2,3,5,4,7,6,11,9,8,10,...
> b ->  1,1,2,1,2,1,4,2,1,1,...
> 
> (amazing!...)  
> 
> Aside from if a -> b implies b -> a, what additionally can be said
> about this transform on sequences?
> 
> thanks,
> Leroy
>  Quet