"Especially-Symmetric" Sequence

[Leroy Quet]

qqquet at mindspring.com (Leroy Quet) wrote in message 
news:<b4be2fdf.0301281541.f91e87 at posting.google.com>...
> Let an "especially-symmetric sequence" be a sequence (or string) of
> terms (numbers, letters, etc.) where, if each a() is a term (specific
> number, letter, etc):
> 
> s(1) = a(1);
> 
> s(m) is either {s(m-1),a(m),s(m-1)}
> or {s(m-1),s(m-1)}
> 
> for all integers m >= 2.
> 
> (s(m) is a concatenation involving s(m-1).)
> 
>.....


It is noteworthy, but somewhat obvious, that a sequence of any length n, 
n = positive
integer, can be obtained, and in exactly one way.

Just take the binary representation of n. Start at left-most (most
significant) one digit.
Moving through the binary representation, from left to right:
*at bit=0, s(m) = {s(m-1),s(m-1)};
*at bit=1, s(m) = {s(m-1),a(m),s(m-1)}.

Thanks,
Leroy Quet